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Theorem 1. \( J\left( {t}_{1}\right) \) is maximum when \( {t}_{1} = {t}_{M} \) and minimum when \( {t}_{1} = {t}_{m} \) . In other words, \( J\left( {t}_{m}\right) \leq J\left( {t}_{1}\right) \leq J\left( {t}_{M}\right) \) for \( {t}_{m} \leq {t}_{1} \leq {t}_{M} \) . | Proof. It follows from (6) and (4) that\n\n\[ J\left( {t}_{1}\right) = r\left( {{t}_{1} + \frac{4\operatorname{Rr}{t}_{1}}{{r}^{2} + {t}_{1}^{2}}}\right) .\n\]\n\nFrom \( \frac{d}{d{t}_{1}}J\left( {t}_{1}\right) = 0 \), we obtain the equation\n\n\[ {t}_{1}^{4} - 2\left( {{2Rr} - {r}^{2}}\right) {t}_{1}^{2} + {4R}{r}^{3... | Yes |
Theorem 1. The circle \( \mathcal{C}\left( {{a}^{\prime },{b}^{\prime }}\right) \) has radius \( \frac{{ab}\left( {{a}^{\prime } + {b}^{\prime }}\right) }{a{a}^{\prime } + b{b}^{\prime } + {a}^{\prime }{b}^{\prime }} \) . | Proof. Let \( x \) be the radius of the circle touching \( \gamma \) internally and also touching \( \alpha \left( {a}^{\prime }\right) \) and \( \beta \left( {b}^{\prime }\right) \) each at a point different from \( O \) . There are two cases in which this circle touches both \( \alpha \left( {a}^{\prime }\right) \) a... | Yes |
Theorem 2. The two centers of similitude \( P\left( {a}^{\prime }\right) \) and \( P\left( {b}^{\prime }\right) \) coincide if and only if\n\n\[ \frac{a}{{a}^{\prime }} + \frac{b}{{b}^{\prime }} = 1 \] | Proof. If the two centers of similitude coincide at the point \( \left( {t,0}\right) \), then by similarity,\n\n\[ {a}^{\prime } : t - {a}^{\prime } = a + b : t - \left( {a - b}\right) = {b}^{\prime } : t + {b}^{\prime }.\]\n\nEliminating \( t \), we obtain (1). The converse is obvious by the uniqueness of the figure. | No |
Theorem 3. The circle \( \mathcal{C}\left( {{a}^{\prime },{b}^{\prime }}\right) \) is an Archimedean circle if and only if \( P\left( {a}^{\prime }\right) \) and \( P\left( {b}^{\prime }\right) \) coincide. | When both \( {a}^{\prime } \) and \( {b}^{\prime } \) are positive, the two centers of similitude \( P\left( {a}^{\prime }\right) \) and \( P\left( {b}^{\prime }\right) \) coincide if and only if the three semicircles \( \alpha \left( {d}^{\prime }\right) ,\beta \left( {b}^{\prime }\right) \) and \( \gamma \) share a c... | Yes |
Theorem 5. If \( T \) is a point on the line \( \mathcal{L} \), then the circle touching the tangents of \( \beta \) through \( T \) with center on the Schoch line \( {\mathcal{L}}_{s} \) is an Archimedean circle. | Proof. Let \( x \) be the radius of this circle. By similarity (see Figure 6),\n\n\[ b + {2r} : b = {2r} - \frac{b - a}{b + a}r : x.\]\n\nFrom this, \( x = r \) . | No |
Theorem 7. For \( n \) satisfying (4), the circle with center on the Schoch line touching \( \alpha \left( {na}\right) \) and \( \beta \left( {nb}\right) \) internally is an Archimedean circle. | Proof. Let \( x \) be the radius of the circle with center given by (2) and touching \( \alpha \left( {na}\right) \) and \( \beta \left( {nb}\right) \) internally, where \( n \) satisfies (4). Since the centers of \( \alpha \left( {na}\right) \) and \( \beta \left( {nb}\right) \) are \( \left( {{na},0}\right) \) and \(... | Yes |
Theorem 8. Let \( {\mathcal{C}}_{1} \) be a circle with center \( O \), passing through a point \( P \) on the \( x \) -axis, and \( {\mathcal{C}}_{2} \) a circle with center on the \( x \) -axis passing through \( O \) . If \( {\mathcal{C}}_{2} \) and the vertical line through \( P \) intersect, then the tangents of \... | Proof. Let \( d \) be the distance between \( O \) and the intersection of the tangent of \( {\mathcal{C}}_{2} \) and the \( x \) -axis, and let \( x \) be the distance between the tangent and \( O \) . We may assume \( {r}_{1} \neq {r}_{2} \) for the radii \( {r}_{1} \) and \( {r}_{2} \) of the circles \( {\mathcal{C}... | Yes |
Theorem 9. Let \( m \) and \( n \) be positive numbers. The Archimedean circles \( \mathcal{A}\left( m\right) \) and \( {\mathcal{A}}^{\prime }\left( n\right) \) coincide if and only if \( m \) and \( n \) satisfy\n\n\[ \frac{1}{ma} + \frac{1}{nb} = \frac{1}{r} = \frac{1}{a} + \frac{1}{b} \] | Proof. By (3) the equations of the tangents \( {t}_{m} \) and \( {t}_{n}^{\prime } \) are\n\n\[ - \left( {{ma} + \left( {m - 2}\right) b}\right) x + 2\sqrt{b\left( {{ma} + \left( {m - 1}\right) b}\right) }y = {2mab}, \]\n\n\[ \left( {{nb} + \left( {n - 2}\right) a}\right) x + 2\sqrt{a\left( {{nb} + \left( {n - 1}\right... | Yes |
Lemma 3. The polar circles of the triangles \( {ABC},{AVW},{BWU},{CUV} \) and the circumcircle of the diagonal triangle are coaxal. The three circles with diameter \( {AU},{BV},{CW} \) are coaxal. The corresponding pencils of circles are orthogonal. | Proof. By \( §{2.3} \), each of the four polar circles is orthogonal to the three circles with diameter \( {AU},{BV},{CW} \) . More over, as each of the quadruples \( \left( {A, U,{B}^{\prime },{C}^{\prime }}\right) \) , \( \left( {B, V,{C}^{\prime },{A}^{\prime }}\right) \) and \( \left( {C, W,{A}^{\prime },{B}^{\prim... | Yes |
Proposition 4. The circumcenter of the diagonal triangle lies on the orthocentric line. | This follows from Lemma 3 and §2.3. | No |
Proposition 6 (Goormaghtigh). The orthopole of a sideline of the complete quadrilateral with respect to the triangle bounded by the three other sidelines lies on the orthocentric line. | Proof. See [1, pp.241-242]. | No |
Proposition 8 (Mention [4]). (1) The following seven circles are members of a pencil \( \Phi \) :\n\n\[ \Gamma \left( {u, v, w}\right) ,\Gamma \left( {u,{v}^{\prime },{w}^{\prime }}\right) ,\Gamma \left( {{u}^{\prime }, v,{w}^{\prime }}\right) ,\Gamma \left( {{u}^{\prime },{v}^{\prime }, w}\right) ,\]\n\nand those with... | This clearly gives Steiner's Theorems 8 and 9. | No |
Proposition 9 (Clawson). The central lines of the pencils \( \Phi \) and \( {\Phi }^{\prime } \) are the common bisectors of the three pairs of lines \( \left( {{FA},{FU}}\right) ,\left( {{FB},{FV}}\right) \), and \( \left( {{FC},{FW}}\right) \) . | Note that, as \( \left( {{FA},{FB}}\right) = \left( {{FV},{FU}}\right) = \left( {{CA},{CB}}\right) \), it is clear that the three pairs of lines \( \left( {{FA},{FU}}\right) ,\left( {{FB},{FV}}\right) ,\left( {{FC},{FW}}\right) \) have a common pair of bisectors \( \left( {f,{f}^{\prime }}\right) \) . These bisectors a... | Yes |
Proposition 11. The locus of the foci of these conics is a circular focal cubic (van Rees focal). | This cubic \( \gamma \) passes through \( A, B, C, U, V, W, F \), the circular points at infinity \( {I}_{\infty },{J}_{\infty } \) and the feet of the altitudes of the diagonal triangle.\n\nThe real asymptote is the image of the Newton line under the homothety \( \mathrm{h}\left( {F,2}\right) \) , and the imaginary as... | Yes |
Proposition 12 (Oppermann). The circles of the pencil generated by the three circles with diameters \( {AU},{BV},{CW} \) are the Monge circle’s of the conics inscribed in the complete quadrilateral. | Proof. See [5, pp.60-61]. | No |
Theorem 1. Given two lines in a plane, let \( A, B, C \) be three points on one line and \( {A}^{\prime },{B}^{\prime },{C}^{\prime } \) three points on the other line. The three points\n\n\[ B{C}^{\prime } \cap C{B}^{\prime },\;C{A}^{\prime } \cap A{C}^{\prime },\;A{B}^{\prime } \cap B{A}^{\prime } \]\n\nare collinear... | Let \( X = B{C}^{\prime } \cap C{B}^{\prime }, Y = C{A}^{\prime } \cap A{C}^{\prime }, Z = A{B}^{\prime } \cap B{A}^{\prime } \) . The points \( {A}^{\prime },{B}^{\prime } \) , \( {C}^{\prime } \) being infinite points, we have \( {CY}\parallel {BZ},{AZ}\parallel {CX} \), and \( {BX}\parallel {AY} \) . See Figure 2. W... | Yes |
Theorem 2 (Steiner). If a, b, c, d are any four lines, the orthocenters of \( \Delta \mathrm{{bcd}} \) , \( \Delta \mathrm{{acd}},\Delta \mathrm{{abd}},\Delta \mathrm{{abc}} \) are collinear. | Proof. Let \( D, E, F \) be the intersections of \( \mathrm{d} \) with \( \mathrm{a},\mathrm{b},\mathrm{c} \), and \( K, L, M, N \) the orthocenters of \( \Delta \mathrm{{bcd}},\Delta \mathrm{{acd}},\Delta \mathrm{{abd}} \), and \( \Delta \mathrm{{abc}} \) . Note that \( K = E\overline{\mathrm{c}} \cap F\overline{\math... | Yes |
Theorem 3. Let \( \bigtriangleup {ABC} \) be a triangle and \( \mathrm{d} \) a line. If \( {A}^{\prime },{B}^{\prime },{C}^{\prime } \) are the pedals of \( A, B, C \) on \( \mathrm{d} \), then the perpendiculars from \( {A}^{\prime },{B}^{\prime },{C}^{\prime } \) to the lines \( {BC},{CA},{AB} \) intersect at one poi... | Proof. Denote by a, b, c the lines \( {BC},{CA},{AB} \) . By Theorem 2, the orthocenters \( K, L, M, N \) of triangles \( \Delta \mathrm{{bcd}},\Delta \mathrm{{acd}},\Delta \mathrm{{abd}},\Delta \mathrm{{abc}} \) lie on a line. Let \( D = \) \( \mathrm{d} \cap \mathrm{a} \), and \( W = {B}^{\prime }\overline{\mathrm{b}... | Yes |
Theorem 5. If \( A, B, C, D \) are four points and \( \mathrm{e} \) is a line, then the orthopoles of \( \mathrm{e} \) with respect to triangles \( \bigtriangleup {BCD},\bigtriangleup {CDA},\bigtriangleup {DAB},\bigtriangleup {ABC} \) are collinear. | Proof. Denote these orthopoles by \( X, Y, Z, W \) respectively. If \( {A}^{\prime },{B}^{\prime },{C}^{\prime },{D}^{\prime } \) are the pedals of \( A, B, C, D \) on e, then \( X = {B}^{\prime }\overline{CD} \cap {C}^{\prime }\overline{BD} \) . Similarly, \( Y = \) \( {C}^{\prime }\overline{AD} \cap {A}^{\prime }\ove... | Yes |
Theorem 6. Given five lines \( \mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d} \), e, the orthopoles of \( \mathrm{e} \) with respect to \( \Delta \mathrm{{bcd}} \) , \( \Delta \) acd, \( \Delta \) abd, \( \Delta \) abc are collinear. | Proof. Denote these orthopoles by \( X, Y, Z, W \) respectively. Let the line d intersect a, b, c at \( D, E, F \), and let \( {D}^{\prime },{E}^{\prime },{F}^{\prime } \) be the pedals of \( D, E, F \) on e.\n\nSince \( E = \mathrm{b} \cap \mathrm{d} \) and \( F = \mathrm{c} \cap \mathrm{d} \) are two vertices of tria... | Yes |
Corollary 2. Let \( {ABCD} \) be a quadrilateral with an incircle \( I\left( r\right) \) tangent to the sides at \( W, X, Y, Z \) . If the excircles \( {I}_{W}\left( {r}_{W}\right) ,{I}_{X}\left( {r}_{X}\right) ,{I}_{Y}\left( {r}_{Y}\right) ,{I}_{Z}\left( {r}_{Z}\right) \) have areas \( {T}_{W},{T}_{X},{T}_{Y},{T}_{Z} ... | Proof. By (1) above, we have \( \frac{{T}_{W}}{{r}_{W}} = \frac{\text{ Area }{XYZ}}{r} \) and \( \frac{{T}_{Y}}{{r}_{Y}} = \frac{\text{ Area }{ZWX}}{r} \) so that\n\n\[ \n\frac{{T}_{W}}{{r}_{W}} + \frac{{T}_{Y}}{{r}_{Y}} = \frac{\operatorname{Area}{XYZ} + \operatorname{Area}{ZWX}}{r} = \frac{T}{r}. \n\] \n\nSimilarly, ... | Yes |
Theorem 1. Let \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) be the intouch triangle of \( {ABC} \), and \( {A}^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime } \) the in-touch triangle of \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) . The triangles \( {A}^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime... | The proof is a simple application of the following cevian nest theorem. \( {}^{1} \) | No |
Theorem 2. Let \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) be the cevian triangle of \( P \) in triangle \( {ABC} \) with homogeneous barycentric coordinates \( \left( {u : v : w}\right) \) with respect to \( {ABC} \), and \( {A}^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime } \) the cevian triangle of \( Q ... | Proof. We compute the absolute barycentric coordinates explicitly.\n\n\[ {A}^{\prime \prime } = \frac{y{B}^{\prime } + z{C}^{\prime }}{y + z} = \frac{y \cdot \frac{{wC} + {uA}}{w + u} + z \cdot \frac{{uA} + {vB}}{u + v}}{y + z} \]\n\n\[ = \frac{\left( {y\left( {u + v}\right) + z\left( {w + u}\right) }\right) {uA} + {zv... | Yes |
Theorem 1. Let \( P \) be a finite point. The line joining \( P \) to its isotomic conjugate if parallel to \( {BC} \) if and only if \( P \) lies on the line through \( A \) parallel to \( {BC} \) or the ellipse through the centroid tangent to \( {AB} \) and \( {AC} \) at \( B \) and \( C \) respectively. In the latte... | Now we consider the possibility for \( P{P}^{\prime } \) not only to be parallel to \( {BC} \), but also equal to one half of its length. This means that the vector \( {PP} \) is \( \pm \frac{1}{2}\left( {C - B}\right) \) . If \( P \) is a finite point on the parallel to \( {BC} \) through \( A \), we write \( P = \lef... | Yes |
Theorem 2. There are four pairs of isotomic conjugates \( P,{P}^{\prime } \) for which the segment \( P{P}^{\prime } \) is parallel to \( {BC} \) and has half of its length. | <table><thead><tr><th>\( i \)</th><th>\( {P}_{i} \)</th><th>\( {P}_{i}^{\prime } \)</th></tr></thead><tr><td>1</td><td>\( \left( {\sqrt{17} - 1 : 4 : - 4}\right) \)</td><td>\( \left( {\sqrt{17} + 1 : 4 : - 4}\right) \)</td></tr><tr><td>2</td><td>\( \left( {\sqrt{17} + 1 : - 4 : 4}\right) \)</td><td>\( \left( {\sqrt{17}... | Yes |
Lemma 2. The orthocenter \( {H}^{\prime } \) of the intouch triangle lies on the line OI. | Proof. Let \( {I}_{1}{I}_{2}{I}_{3} \) be the excentral triangle. The lines \( {YZ} \) and \( {I}_{2}{I}_{3} \) are parallel because both are perpendicular to \( {AI} \) . Similarly, \( {ZX}//{I}_{3}{I}_{1} \) and \( {XY}//{I}_{1}{I}_{2} \) . See Figure 2. Hence, the excentral triangle and the intouch triangle are homo... | Yes |
Corollary 3. The line joining \( {A}_{1} \) to the projection of \( X \) on \( {YZ} \) passes through the midpoint of the bisector of angle \( A \) . | Proof. In Figure 3, \( {X}^{\prime }X \) is parallel to the bisector of angle \( A \) and its midpoint is the projection of \( X \) on \( {YZ} \) . | No |
Theorem 4. The triangle \( {ABC} \) is isosceles if and only if triangles from some element of \( \tau \) have the same perimeter. | Proof. This time there are only two representative cases.\n\nCase 1: \( \left( {{G}_{a}^{ - },{G}_{a}^{ + }}\right) \) . By assumption,\n\n\[ p\left( {G}_{a}^{ - }\right) - p\left( {G}_{a}^{ + }\right) = \frac{\sqrt{2{a}^{2} - {b}^{2} + 2{c}^{2}}}{3} - \frac{\sqrt{2{a}^{2} + 2{b}^{2} - {c}^{2}}}{3} = 0. \]\n\nWhen we m... | Yes |
Theorem 5. The centroids \( {G}_{{G}_{a}^{ - }},{G}_{{G}_{a}^{ + }},{G}_{{G}_{b}^{ - }},{G}_{{G}_{b}^{ + }},{G}_{{G}_{c}^{ - }},{G}_{{G}_{c}^{ + }} \) of the triangles from \( {\sigma }_{G} \) lie on the image of the Steiner ellipse of \( {ABC} \) under the homothety \( \mathrm{h}\left( {G,\frac{\sqrt{7}}{6}}\right) \)... | Proof. We look for the conic through five of the centroids and check that the the sixth centroid lies on it. The trilinear coordinates of \( {G}_{{G}_{a}^{ - }} \) are \( \frac{2}{a} : \frac{11}{b} : \frac{5}{c} \) while those of other centroids are similar. It follows that they all lie on the ellipse with the equation... | Yes |
Theorem 6. The circumcenters \( {O}_{{G}_{a}^{ - }},{O}_{{G}_{a}^{ + }},{O}_{{G}_{b}^{ - }},{O}_{{G}_{b}^{ + }},{O}_{{G}_{c}^{ - }},{O}_{{G}_{c}^{ + }} \) of the triangles from \( {\sigma }_{G} \) lie on the circle whose center \( {O}_{G} \) is a central point with the first trilinear coordinate \[ \frac{{10}{a}^{4} - ... | Proof. The proof is conceptually simple but technically involved so that we shall only outline how it could be done on a computer. In order to find points \( {O}_{{G}_{a}^{ - }},{O}_{{G}_{a}^{ + }} \) , \( {O}_{{G}_{b}^{ - }},{O}_{{G}_{b}^{ + }},{O}_{{G}_{c}^{ - }},{O}_{{G}_{c}^{ + }} \) we use the circumcenter functio... | Yes |
Theorem 8. (1) The triangles \( {O}_{{G}_{a}^{ - }}{O}_{{G}_{b}^{ - }}{O}_{{G}_{c}^{ - }} \) and \( {O}_{{G}_{b}^{ + }}{O}_{{G}_{c}^{ + }}{O}_{{G}_{a}^{ + }} \) are congruent. They are orthologic to \( {BCA} \) and \( {CAB} \), respectively. | Proof. (1) The points \( {O}_{{G}_{a}^{ - }} \) and \( {O}_{{G}_{a}^{ + }} \) have trilinear coordinates\n\n\[ a\left( {5{c}^{2} - {a}^{2} - {b}^{2}}\right) : \frac{{2h}\left( {3,3,5,2,2,1}\right) }{b} : \frac{h\left( {6,1,3,1,2,4}\right) }{c}, \]\n\n\[ a\left( {5{b}^{2} - {a}^{2} - {c}^{2}}\right) : \frac{h\left( {6,3... | Yes |
Theorem 9. (1) The relation \( b = c \) holds in \( {ABC} \) if and only if \( {O}_{{G}_{a}^{ - }} \) is on \( {BG} \) and/or \( {O}_{{G}_{a}^{ + }} \) is on \( {CG} \) . | Proof. (1) for \( {O}_{{G}_{a}^{ - }} \) . Since the trilinear coordinates of \( {O}_{{G}_{a}^{ - }}, G \) and \( B \) are\n\n\[ a\left( {5{c}^{2} - {a}^{2} - {b}^{2}}\right) : \frac{{2h}\left( {3,3,5,2,2,1}\right) }{b} : \frac{h\left( {6,1,3,1,2,4}\right) }{c}, \]\n\n\( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \) and \... | No |
Theorem 10. The triangles \( {T}_{{G}_{a}^{ - }}{T}_{{G}_{b}^{ - }}{T}_{{G}_{c}^{ - }} \) and \( {T}_{{G}_{a}^{ + }}{T}_{{G}_{b}^{ + }}{T}_{{G}_{c}^{ + }} \) are directly similar to each other or to \( {ABC} \) if and only if \( {ABC} \) is equilateral. | Proof. For \( {T}_{{G}_{a}^{ - }}{T}_{{G}_{b}^{ - }}{T}_{{G}_{c}^{ - }} \) and \( {T}_{{G}_{a}^{ + }}{T}_{{G}_{b}^{ + }}{T}_{{G}_{c}^{ + }} \) . The point \( {T}_{{G}_{a}^{ - }} \) has \( \frac{{p}_{1}}{a} : \frac{{p}_{2}}{b} : \frac{{p}_{3}}{c} \) as trilinear coordinates, where \[ {p}_{1} = 3{a}^{2}\left( {{a}^{2} + ... | Yes |
Theorem 11. (1) \( {T}_{{G}_{a}^{ - }}{T}_{{G}_{b}^{ - }}{T}_{{G}_{c}^{ - }} \) and \( {T}_{{G}_{a}^{ + }}{T}_{{G}_{b}^{ + }}{T}_{{G}_{c}^{ + }} \) are orthologic to \( {ABC} \) if and only if \( k = - \frac{3}{2} \) . | Proof. All parts have similar proofs. For example, in the first, we find that the triangles \( {T}_{{G}_{a}^{ - }}{T}_{{G}_{b}^{ - }}{T}_{{G}_{c}^{ - }} \) and \( {ABC} \) are orthologic if and only if \( - \frac{\left( {{a}^{2} + {b}^{2} + {c}^{2}}\right) \left( {{2k} + 3}\right) }{{12}\left( {k + 1}\right) } = 0 \) . | No |
Theorem 12. The triangles \( {H}_{{G}_{a}^{ - }}{H}_{{G}_{b}^{ - }}{H}_{{G}_{c}^{ - }} \) and \( {H}_{{G}_{a}^{ + }}{H}_{{G}_{b}^{ + }}{H}_{{G}_{c}^{ + }} \) are orthologic if and only if \( {ABC} \) is an equilateral triangle. | Proof. Substituting the coordinates of \( {H}_{{G}_{a}^{ - }},{H}_{{G}_{b}^{ - }},{H}_{{G}_{c}^{ - }},{H}_{{G}_{a}^{ + }},{H}_{{G}_{b}^{ + }},{H}_{{G}_{c}^{ + }} \) into the condition for triangles to be orthologic (see the proof of Theorem 6), we obtain\n\n\[ \frac{\left( {{a}^{2} + {b}^{2} + {c}^{2}}\right) \left\lbr... | Yes |
Theorem 13. The triangles \( {F}_{{G}_{a}^{ - }}{F}_{{G}_{b}^{ - }}{F}_{{G}_{c}^{ - }} \) and \( {F}_{{G}_{a}^{ + }}{F}_{{G}_{b}^{ + }}{F}_{{G}_{c}^{ + }} \) have the same Brocard angle and area. The triangle \( {ABC} \) is equilateral if and only if this area is \( \frac{3}{16} \) of the area of \( {ABC} \) . | Proof. Recall the formula \( \frac{1}{2}\left| {{x}_{1}\left( {{y}_{2} - {y}_{3}}\right) + {x}_{2}\left( {{y}_{3} - {y}_{1}}\right) + {x}_{3}\left( {{y}_{1} - {y}_{2}}\right) }\right| \) for the area of the triangle with vertices \( \left( {{x}_{1},{y}_{1}}\right) ,\left( {{x}_{2},{y}_{2}}\right) ,\left( {{x}_{3},{y}_{... | No |
Theorem 14. The triangles \( {K}_{{G}_{a}^{ - }}{K}_{{G}_{b}^{ - }}{K}_{{G}_{c}^{ - }} \) and \( {K}_{{G}_{a}^{ + }}{K}_{{G}_{b}^{ + }}{K}_{{G}_{c}^{ + }} \) have the area equal to \( \frac{7}{64} \) of the area of \( {ABC} \) if and only if \( {ABC} \) is an equilateral triangle. | Proof. The difference \( \left| {{K}_{{G}_{a}^{ - }}{K}_{{G}_{b}^{ - }}{K}_{{G}_{c}^{ - }}}\right| - \frac{7}{64}\left| {ABC}\right| \) is equal to\n\n\[ \frac{3\Delta T}{{64}\left( {5{b}^{2} + 8{c}^{2} - {a}^{2}}\right) \left( {5{c}^{2} + 8{a}^{2} - {b}^{2}}\right) \left( {5{a}^{2} + 8{b}^{2} - {c}^{2}}\right) },\]\n\... | Yes |
Theorem 15. The triangles \( {K}_{{G}_{a}^{ - }}{K}_{{G}_{b}^{ - }}{K}_{{G}_{c}^{ - }} \) and \( {K}_{{G}_{a}^{ + }}{K}_{{G}_{b}^{ + }}{K}_{{G}_{c}^{ + }} \) have the same area if and only if the triangle \( {ABC} \) is isosceles. | Proof. The difference \( \left| {{K}_{{G}_{a}^{ - }}{K}_{{G}_{b}^{ - }}{K}_{{G}_{c}^{ - }}}\right| - \left| {{K}_{{G}_{a}^{ + }}{K}_{{G}_{b}^{ + }}{K}_{{G}_{c}^{ + }}}\right| \) is equal to\n\n\[ \frac{{81\Delta }\left( {b - c}\right) \left( {b + c}\right) \left( {c - a}\right) \left( {c + a}\right) \left( {a - b}\righ... | Yes |
Theorem 16. The triangles \( {L}_{{G}_{a}^{ - }}{L}_{{G}_{b}^{ - }}{L}_{{G}_{c}^{ - }} \) and \( {L}_{{G}_{a}^{ + }}{L}_{{G}_{b}^{ + }}{L}_{{G}_{c}^{ + }} \) have the same areas and Brocard angles. This area is equal to \( \frac{3}{4} \) of the area of \( {ABC} \) and/or this Brocard angle is equal to the Brocard angle... | Proof. The common area is \( \frac{h\left( {{10},{10},{10},1,1,1}\right) }{112\Delta } \) while the tangent of the common Brocard angle is \( \frac{h\left( {{10},{10},{10},1,1,1}\right) }{{4\Delta }{p}_{2}\left( {ABC}\right) h\left( {2,2,2, - 7, - 7, - 7}\right) } \) . It follows that the difference\n\n\[ \n\frac{3}{4}... | Yes |
Lemma 2. (a) If \( a < \frac{R}{2},\sin \frac{{\theta }_{aa}}{2} = \frac{a}{R - a} \) . (See Figure 2A). | Proof. These are clear from Figures 2A and 2B. | No |
Lemma 3. If \( a \) and \( b \) are unequal and each \( < \frac{R}{2} \), then \( {\theta }_{aa} + {\theta }_{bb} > 2{\theta }_{ab} \). | Proof. In Figure 1A, consider angle \( {AOP} \), where \( P \) is a point on the circle \( \left( A\right) \). The angle \( {AOP} \), is maximum when line \( {OP} \) is tangent to the circle \( \left( A\right) \). This maximum is \( \frac{{\theta }_{aa}}{2} \geq \angle {AOQ} \), where \( Q \) is the point of tangency o... | No |
Corollary 4. If \( a, b, c \) are not the same, then \( {\theta }_{aa} + {\theta }_{bb} + {\theta }_{cc} > {\theta }_{ab} + {\theta }_{bc} + {\theta }_{ca} \) . | Proof. Write\n\n\[ \n{\theta }_{aa} + {\theta }_{bb} + {\theta }_{cc} = \frac{{\theta }_{aa} + {\theta }_{bb}}{2} + \frac{{\theta }_{bb} + {\theta }_{cc}}{2} + \frac{{\theta }_{cc} + {\theta }_{aa}}{2} \n\] \n\nand apply Lemma 3. | No |
Lemma 5. If three circles of radii \( x, z, z \) are tangent externally to each other, and are each tangent internally to a circle of radius \( R \), then\n\n\[ z = \frac{{4Rx}\left( {R - x}\right) }{{\left( R + x\right) }^{2}}. \] | Proof. By the Descartes circle theorem [2], we have\n\n\[ 2\left( {\frac{1}{{R}^{2}} + \frac{1}{{x}^{2}} + \frac{2}{{z}^{2}}}\right) = {\left( -\frac{1}{R} + \frac{1}{x} + \frac{2}{z}\right) }^{2}, \]\n\nfrom which the result follows. | No |
Theorem 6. For a given \( R \), a nonoverlapping arrangement of pattern \( {E}_{3}\left( {abcacb}\right) \) or \( {E}_{4}\left( {bcabac}\right) \) with \( a \leq b \leq c \) and \( a + b + c = R \) exists if \( {\gamma R} \leq a \leq \frac{1}{3}R \), where\n\n\[ \gamma = \frac{1 + \sqrt[3]{{19} + {12}\sqrt{87}} + \sqrt... | Proof. For \( b = a \) and the largest \( c = R - {2a} \) for a nonoverlapping arrangement \( {\mathrm{E}}_{3}\left( {abcacb}\right) \), Lemma 5 gives\n\n\[ \frac{{4Ra}\left( {R - a}\right) }{{\left( R + a\right) }^{2}} - \left( {R - {2a}}\right) = \frac{f\left( \frac{a}{R}\right) {R}^{3}}{{\left( R + a\right) }^{2}} =... | Yes |
Corollary 7. The sufficient condition \( {\gamma R} \leq a \leq \frac{1}{3}R \) also applies to patterns \( {E}_{1} \) and \( {E}_{2} \) . | Outside the range \( {\gamma R} \leq a \leq \frac{R}{3} \), patterns \( {\mathrm{E}}_{3}\left( {abcacb}\right) \) and \( {\mathrm{E}}_{4}\left( {bcabac}\right) \) still can have nonoverlapping circles. Both of the patterns involve Figure 5 and \( z = \) \( \frac{{4Rx}\left( {R - x}\right) }{{\left( R + x\right) }^{2}} ... | Yes |
Lemma 1. Let \( P = \left( {{a}^{2}{vw} : {b}^{2}{wu} : {c}^{2}{uv}}\right) \) be a point on the circumcircle (so that \( u + v + w = 0) \) . For a point \( Q = \left( {x : y : z}\right) \) different from \( P \) and not lying on the circumcircle, the line \( {PQ} \) intersects the circumcircle again at the point \( \l... | Proof. Entering the coordinates\n\n\[ \left( {\mathbb{X},\mathbb{Y},\mathbb{Z}}\right) = \left( {{a}^{2}{vw} + {tx} : {b}^{2}{wu} + {ty} : {c}^{2}{uv} + {tz}}\right) \]\n\ninto the equation of the circumcircle\n\n\[ {a}^{2}\mathbb{{YZ}} + {b}^{2}\mathbb{Z}\mathbb{X} + {c}^{2}\mathbb{X}\mathbb{Y} = 0 \]\n\nwe obtain\n\n... | Yes |
Proposition 3. Let \( {X}^{\prime \prime },{Y}^{\prime \prime },{Z}^{\prime \prime } \) be the second intersections of the circumcircle with the lines \( {DX},{EY},{FZ} \) respectively. The lines \( A{X}^{\prime \prime }, B{Y}^{\prime \prime }, C{Z}^{\prime \prime } \) bound the anticevian triangle of \( {X}_{57} \) . | Proof. By Lemma 1, these are the points\n\n\[ \n{X}^{\prime \prime } = \left( {\frac{{a}^{2}}{s - a} : \frac{b\left( {b - c}\right) }{s - b} : \frac{c\left( {c - b}\right) }{s - c}}\right) , \n\]\n\n\[ \n{Y}^{\prime \prime } = \left( {\frac{a\left( {a - c}\right) }{s - a} : \frac{{b}^{2}}{s - b} : \frac{c\left( {c - a}... | Yes |
Proposition 4. \( {X}_{57} \) is the perspector of the triangle bounded by the polars of \( A \) , \( B, C \) with respect to the circle through the excenters. | Proof. As is easily verified, the equation of the circumcircle of the excentral triangle is\n\n\[ \n{a}^{2}{yz} + {b}^{2}{zx} + {c}^{2}{xy} + \left( {x + y + z}\right) \left( {{bcx} + {cay} + {abz}}\right) = 0.\n\]\n\nThe polars are the lines\n\n\[ \n\frac{x}{s} + \frac{y}{b} + \frac{z}{c} = 0 \n\]\n\n\[ \n\begin{matri... | Yes |
Lemma 8. Consider a triangle \( {ABC} \) with intouch triangle \( {XYZ} \), and a line \( \mathcal{L} \) intersecting the sides \( {BC},{CA},{AB} \) at \( {A}^{\prime },{B}^{\prime },{C}^{\prime } \) respectively. The line \( \mathcal{L} \) is tangent to the incircle if and only if one of the following conditions holds... | Proof. Let \( {A}^{\prime }{B}^{\prime } \) be a tangent to the incircle. By Brianchon’s theorem applied to the circumscribed hexagon \( {AY}{B}^{\prime }{A}^{\prime }{XB} \) it immediately follows that \( A{A}^{\prime },{YX} \) and \( {B}^{\prime }B \) are concurrent.\n\nNow suppose \( A{A}^{\prime },{YX} \) and \( {B... | No |
Theorem 1. If triangles \( {ABC} \) and \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) are orthologic with centers \( P,{P}^{\prime } \) then the barycentric coordinates of \( P \) with respect to \( {ABC} \) are equal to the barycentric coordinates of \( {P}^{\prime } \) with respect to \( {A}^{\prime }{B}^{\prime }{C}... | Proof. Since \( {A}^{\prime }{P}^{\prime },{B}^{\prime }{P}^{\prime },{C}^{\prime }{P}^{\prime } \) are perpendicular to \( {BC},{CA},{AB} \) respectively, we have\n\n\[ \sin {B}^{\prime }{P}^{\prime }{C}^{\prime } = \sin A,\;\sin {P}^{\prime }{B}^{\prime }{C}^{\prime } = \sin {PAC},\;\sin {P}^{\prime }{C}^{\prime }{B}... | Yes |
Proposition 2. If \( P \) lies on the circumcircle, the line joining \( P \) to \( Q \) always passes through the deLongchamps point \( {X}_{20} \) . | Proof. The equation of the line \( {PQ} \) is\n\n\[ \mathop{\sum }\limits_{\text{cyclic }}\left( {{S}_{B} - {S}_{C}}\right) \left( {{S}_{A} + t}\right) \left( {{S}_{A}^{3}{\left( {S}_{B} - {S}_{C}\right) }^{2}}\right.\n\n\[ + \left( {{S}_{B} + {S}_{C} + {2t}}\right) \left( {{S}_{AA}\left( {{S}_{BB} - {S}_{BC} + {S}_{CC... | No |
Proposition 1. The pedal of the real conic \( \kappa \) has a node, cusp or acnode depending on whether \( S \) is outside, on, or inside \( \kappa \) . | Proof. By the calculation of the second degree terms of \( G \), the singular tangents at the point \( S \) of the pedal are the perpendiculars to the two tangents from \( S \) to the conic \( \kappa \) . Thus the type of node depends on the position of \( S \) with respect to the conic since that determines how \( {G}... | No |
Proposition 2. A real quartic curve has the equation \( G = A{\left( {x}^{2} + {y}^{2}\right) }^{2} + \left( {{x}^{2} + }\right. \) \( \left. {y}^{2}\right) \left( {{Bx} + {Cy}}\right) + D{x}^{2} + {Exy} + F{y}^{2} = 0 \) for \( A \neq 0 \) iff it is bicircular with double point at the origin. Thus the pedal of an elli... | Proof. A quartic has a double point at the origin iff there are no terms of degree less than 2 in the (inhomogeneous) equation \( G = 0 \) . There are double points at the circular points iff \( G\left( {x, y, z}\right) \) vanishes to second order when evaluated at the circular points; hence iff the gradient of \( G \)... | Yes |
Proposition 3. A bicircular quartic is the pedal of an ellipse or hyperbola. | Proof. Using the equation for the pedal of a conic as in Section 2 we consider the system of equations \( A = {4ac} - {b}^{2}, B = {4cd} - {2be}, C = {4ae} - {2bdy}, D = {4cf} - {e}^{2} \) , \( E = {2ed} - {4bf}, F = {4af} - {d}^{2} \) . One can easily see that this is equivalent to a (symmetric) matrix equation \( Y =... | Yes |
Proposition 4. A singular circular cubic with singularity at the origin has an equation \( G = \left( {{x}^{2} + {y}^{2}}\right) \left( {{Bx} + {Cy}}\right) + D{x}^{2} + {Exy} + F{y}^{2} = 0 \) and conversely. This is the pedal of a parabola. | Proof. The cubic is singular at the origin iff there are no terms of degree less than two; the curve is circular iff the cubic terms vanish at the circular points iff \( {x}^{2} + {y}^{2} \) is a factor of the cubic terms. | No |
Lemma 1. If the centroid \( G \) of the triangle \( {ABC} \) is inside the incircle \( \left( I\right) \), then\n\n\[{a}^{2} < {4bc},\;{b}^{2} < {4ca},\;{c}^{2} < {4ab}.\] | Proof. Because \( G \) is inside \( \left( I\right) \), we have \( {\overrightarrow{IG}}^{2} \leq {r}^{2},{\left( \overrightarrow{AG} - \overrightarrow{AI}\right) }^{2} \leq {r}^{2},{\overrightarrow{AG}}^{2} + \n\n\( {\overrightarrow{AI}}^{2} - 2\overrightarrow{AG} \cdot \overrightarrow{AI} \leq {r}^{2} \) . This inequ... | Yes |
Proposition 1 ([6,§2]). The points \( {P}_{a},{P}_{b},{P}_{c} \) are collinear if and only if \( {t}^{2} + {Rt} + \) \( \frac{1}{2}{Rr} = 0 \), i.e., | \[ t = \frac{R \pm \sqrt{{R}^{2} - {2Rr}}}{2} = \frac{R \pm {OI}}{2}. \] | Yes |
Proposition 2. The triangles \( {ABC} \) and \( {P}_{a}{P}_{b}{P}_{c} \) are perspective if and only if\n\n(1) \( t = 0 : {P}_{a}{P}_{b}{P}_{c} \) is the medial triangle, or\n\n(2) \( t = - r : {P}_{a},{P}_{b},{P}_{c} \) are the projections of the incenter \( I = {X}_{1} \) on the perpendicular bisectors. | In the latter case, \( {P}_{a},{P}_{b},{P}_{c} \) obviously lie on the circle with diameter \( {IO} \) . The two triangles are indirectly similar and their perspector is \( {X}_{8} \) (Nagel point). | No |
Proposition 3. The Mandart triangle \( {\mathbf{T}}_{t} \) and the medial triangle \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) have the same area if and only if either :\n\n(1) \( t = 0 : {\mathbf{T}}_{t} \) is the medial triangle,\n\n(2) \( t = - R \),\n\n(3) \( t \) is solution of: \( {t}^{2} + {Rt} + {Rr} = 0 \) . | This equation has two distinct (real) solutions when \( R > {4r} \), hence there are three Mandart triangles, distinct of \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \), having the same area as \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) . See Figure 2. In the very particular situation \( R = {4r} \), the equation giv... | Yes |
Proposition 5 ([6,§7]). The Mandart triangle \( {\mathbf{T}}_{t} \) and the medial triangle are perspective at \( O \) . As \( t \) varies, the perspectrix envelopes the parabola \( {\mathcal{P}}_{M} \) with focus \( {X}_{124} \) and directrix \( {X}_{3}{X}_{10} \) . | We call \( {\mathcal{P}}_{M} \) the Mandart parabola. It has equation\n\n\[ \mathop{\sum }\limits_{\text{cyclic }}\frac{{x}^{2}}{\left( {b - c}\right) \left( {b + c - a}\right) } = 0. \]\n\nTriangle \( {ABC} \) is clearly self-polar with respect to \( {\mathcal{P}}_{M} \) . The directrix is the line \( {X}_{3}{X}_{10} ... | Yes |
Proposition 6 ([5,2, p.551]). The Mandart triangle \( {\mathbf{T}}_{t} \) and \( {ABC} \) are orthologic. The perpendiculars from \( A, B, C \) to the corresponding sidelines of \( {P}_{a}{P}_{b}{P}_{c} \) are concurrent at | \[ {Q}_{t} = \left( {\frac{a}{a{S}_{A} + {4\Delta t}} : \cdots : \cdots }\right) . \]\n\nAs \( t \) varies, the locus of \( {Q}_{t} \) is the Feuerbach hyperbola. | Yes |
Proposition 7. [6, \( §§1,2 \) ] The Mandart triangle \( {\mathbf{T}}_{t} \) and the extouch triangle are orthologic. The perpendiculars drawn from \( {A}_{1},{B}_{1},{C}_{1} \) to the corresponding sidelines of \( {\mathbf{T}}_{t} = {P}_{a}{P}_{b}{P}_{c} \) are concurrent at \( S \) . As \( t \) varies, the locus of \... | We call \( {\mathcal{H}}_{M} \) the Mandart hyperbola. It has equation\n\n\[ \mathop{\sum }\limits_{\text{cyclic }}\left( {b - c}\right) \left\lbrack {\left( {c + a - b}\right) \left( {a + b - c}\right) {x}^{2} + {\left( b + c - a\right) }^{2}{yz}}\right\rbrack = 0 \] | Yes |
Proposition 10. \( Z \) is a point on the nine-point circle and \( {Z}^{\prime } \) is the foot of the fourth normal drawn from \( N \) to \( {\gamma }_{M} \) . | Proof. The lines \( N{M}_{a}, N{M}_{b}, N{M}_{c} \) are indeed already three such normals hence \( {\Gamma }_{M} \) is the Joachimsthal circle of \( N \) with respect to \( {\gamma }_{M} \). This yields that \( {\Gamma }_{M} \) must pass through the reflection in \( {\omega }_{M} \) of the foot of the fourth normal. Se... | No |
Proposition 11. The points \( {M}_{a},{M}_{b},{M}_{c}, M, N,{\omega }_{M} \) and \( {Z}^{\prime } \) lie on a same rectangular hyperbola whose asymptotes are parallel to the axes of \( {\gamma }_{M} \) . | Proof. This hyperbola is the Apollonian hyperbola of \( N \) with respect to \( {\gamma }_{M} \) . | Yes |
Proposition 14. The triangles \( {ABC} \) and \( {P}_{a}{P}_{b}{P}_{c} \) are perspective if and only if \( k \) is solution of:\n\n\[ \n{\Psi }_{2}\left( {u, v, w}\right) {t}^{2} + {\Psi }_{1}\left( {u, v, w}\right) t + {\Psi }_{0}\left( {u, v, w}\right) = 0 \n\] | where :\n\n\[ \n{\Psi }_{2}\left( {u, v, w}\right) = - \frac{1}{2}{abc}\left( {a + b + c}\right) {\left( u + v + w\right) }^{2}\mathop{\sum }\limits_{\text{cyclic }}\left( {b - c}\right) \left( {b + c - a}\right) {S}_{A}u, \n\]\n\n\[ \n{\Psi }_{1}\left( {u, v, w}\right) = \frac{1}{2}\left( {a + b + c}\right) \left( {u ... | Yes |
Lemma 15. For a given \( P \) and a corresponding Mandart triangle \( {\mathbf{T}}_{t}\left( P\right) = \) \( {P}_{a}{P}_{b}{P}_{c} \), the locus of \( {R}_{a} = B{P}_{b} \cap C{P}_{c} \), when \( t \) varies, is a conic \( {\gamma }_{a} \) . | Proof. The correspondence on the pencils of lines with poles \( B \) and \( C \) mapping the lines \( B{P}_{b} \) and \( C{P}_{c} \) is clearly an involution. Hence, the common point of the two lines must lie on a conic. | Yes |
Lemma 16. The three conics \( {\gamma }_{a},{\gamma }_{b},{\gamma }_{c} \) have three points in common: \( H \) and the (not always real) sought perspectors \( {R}_{1} \) and \( {R}_{2} \) . Their jacobian must degenerate | into three lines, one always real \( {\mathcal{L}}_{P} \) containing \( {R}_{1} \) and \( {R}_{2} \), two other passing through \( H \) . | Yes |
Lemma 17. \( {\mathcal{L}}_{P} \) contains the Nagel point \( {X}_{8} \) . In other words, \( {X}_{8},{R}_{1} \) and \( {R}_{2} \) are always collinear. | With \( P = \left( {u : v : w}\right) ,{\mathcal{L}}_{P} \) has equation :\n\n\[ \mathop{\sum }\limits_{\text{cyclic }}\frac{a\left( {{cv} - {bw}}\right) }{b + c - a}x = 0 \] | No |
Corollary 18. When \( P \) lies on \( {IH} \), there is only one (always real) Mandart triangle \( {\mathbf{T}}_{t}\left( P\right) \) perspective to \( {ABC} \) . The perspector \( R \) is the intersection of the lines \( H{X}_{8} \) and \( P{X}_{78} \) . | Proof. This is obvious since equation (2) is at most of the first degree when \( P \) lies on \( {IH} \) . | No |
Corollary 19. When \( P \) (different from \( I \) and \( H \) ) lies on the conic seen above, there are two (not always real) Mandart triangles \( {\mathbf{T}}_{t}\left( P\right) \) perspective to \( {ABC} \) obtained for two opposite values \( {t}_{1} \) and \( {t}_{2} \) . The vertices of the triangles are therefore... | In the figure 18, we have taken \( P = {X}_{500} \) (orthocenter of the incentral triangle).\n\n\n\nFigure 18. Two triangles \( {P}_{a}{P}_{b}{P}_{c} \) perspective with \( {ABC} \) having vertices symmetric in the s... | Yes |
Corollary 20. When \( P \) (different from \( I, H,{X}_{1490} \) ) lies on the Darboux cubic, there are two (always real) Mandart triangles \( {\mathbf{T}}_{t}\left( P\right) \) perspective to \( {ABC} \), one of them being the pedal triangle of \( P \) with a perspector on the Lucas cubic. | Since one perspector, say \( {R}_{1} \), is known, the construction of the other is simple: it is the \ | No |
Proposition 21. The triangles \( {A}^{\prime }{B}^{\prime }{C}^{\prime } \) and \( {P}_{a}{P}_{b}{P}_{c} \) have the same area if and only if\n\n(1) \( t = 0 \), or\n\n(2) \( t = - \frac{{bc}\left( {b + c}\right) u + {ca}\left( {c + a}\right) v + {ab}\left( {a + b}\right) w}{{2R}\left( {a + b + c}\right) \left( {u + v ... | \[ {}^{10}{abc}\left( {a + b + c}\right) {\left( u + v + w\right) }^{2}{t}^{2} + {2\Delta }\left( {u + v + w}\right) \left( {\mathop{\sum }\limits_{\text{cyclic }}{bc}\left( {b + c}\right) u}\right) t + 8{\Delta }^{2}\left( {{a}^{2}{vw} + }\right. \]\n\[ \left. {{b}^{2}{wu} + {c}^{2}{uv}}\right) = 0. \] | Yes |
Proposition 22. As \( t \) varies, each line \( {P}_{b}{P}_{c},{P}_{c}{P}_{a},{P}_{a}{P}_{b} \) still envelopes a parabola. | Denote these parabolas by \( {\mathcal{P}}_{a},{\mathcal{P}}_{b},{\mathcal{P}}_{c} \) respectively. \( {\mathcal{P}}_{a} \) has focus the projection \( {F}_{a} \) of \( P \) on \( {AI} \) and directrix \( {\ell }_{a} \) parallel to \( {AI} \) at \( {E}_{a} \) such that \( \overrightarrow{P{E}_{a}} = \cos A\overrightarr... | No |
Proposition 24. The Mandart triangle \( {\mathbf{T}}_{t}\left( P\right) \) and \( {ABC} \) are orthologic. The perpendiculars from \( A, B, C \) to the corresponding sidelines of \( {P}_{a}{P}_{b}{P}_{c} \) are concurrent at \( Q = \left( {\frac{{a}^{2}}{{at} + {2\Delta u}} : \cdots : \cdots }\right) \) . As \( t \) va... | This conic has equation\n\n\[ \mathop{\sum }\limits_{\text{cyclic }}{a}^{2}\left( {{cv} - {bw}}\right) {yz} = 0 \]\n\nIt is tangent at \( I \) to \( {IP} \), and is a rectangular hyperbola if and only if \( P \) lies on the line \( {OI}\left( {P \neq I}\right) \) . When \( P = I \), the triangles are homothetic at \( I... | Yes |
Theorem 1.\n\n\[ \n{\lambda }_{1}{R}_{1}^{t} + {\lambda }_{2}{R}_{2}^{t} + {\lambda }_{3}{R}_{3}^{t} \geq {2}^{t}\sqrt{{\lambda }_{1}{\lambda }_{2}{\lambda }_{3}}\left( {\frac{{r}_{1}^{t}}{\sqrt{{\lambda }_{1}}} + \frac{{r}_{2}^{t}}{\sqrt{{\lambda }_{2}}} + \frac{{r}_{3}^{t}}{\sqrt{{\lambda }_{3}}}}\right) .\n\]\n\n(2)... | Proof. As for instance in [1] we have\n\n\[ \n{R}_{1} \geq \frac{{a}_{3}}{{a}_{1}}{r}_{2} + \frac{{a}_{2}}{{a}_{1}}{r}_{3},\;{R}_{2} \geq \frac{{a}_{1}}{{a}_{2}}{r}_{3} + \frac{{a}_{3}}{{a}_{2}}{r}_{1},\;{R}_{3} \geq \frac{{a}_{2}}{{a}_{3}}{r}_{1} + \frac{{a}_{1}}{{a}_{3}}{r}_{2}.\n\]\n\nUsing the power means inequalit... | Yes |
\[ \mathop{\sum }\limits_{{i = 1}}^{3}\frac{{\lambda }_{i}}{{r}_{i}^{t}} \geq {2}^{t}\sqrt{{\lambda }_{1}{\lambda }_{2}{\lambda }_{3}}\mathop{\sum }\limits_{{i = 1}}^{3}\frac{1}{\sqrt{{\lambda }_{i}}{R}_{i}^{t}}, \] | The proofs of these inequalities follow from Theorem 1 upon application of transformations such as\n\n(i) inversion with respect to the circle \( \mathcal{C}\left( {P,\sqrt{{R}_{1}{R}_{2}{R}_{3}}}\right) \) resulting in \( {R}_{i} \mapsto \frac{{R}_{1}{R}_{2}{R}_{3}}{{R}_{i}} \) and \( {r}_{i} \mapsto {R}_{i}{r}_{i} \)... | No |
Proposition 1. The triangle \( {X}_{\varepsilon }{Y}_{\varepsilon }{Z}_{\varepsilon } \) and \( {ABC} \) perspective at the Vecten point | \[ {V}_{\varepsilon } = \left( {\frac{1}{{S}_{A} + {\varepsilon S}} : \frac{1}{{S}_{B} + {\varepsilon S}} : \frac{1}{{S}_{C} + {\varepsilon S}}}\right) . \] | Yes |
Proposition 2. (b) For \( \varepsilon = \pm 1 \), the points \( {A}_{\varepsilon },{B}_{\varepsilon } \) and \( {C}_{\varepsilon } \) are collinear: The line containing them is parallel to the orthic axis. | Proof. The line containing the points \( {A}_{\varepsilon },{B}_{\varepsilon } \) and \( {C}_{\varepsilon } \) has equation\n\n\[ \left( {{S}_{A} + {\varepsilon S}}\right) x + \left( {{S}_{B} + {\varepsilon S}}\right) y + \left( {{S}_{C} + {\varepsilon S}}\right) z = 0. \] | Yes |
Proposition 3. (a) The centers \( X,{Y}_{\varepsilon },{Z}_{\varepsilon } \) of the squares \( {\operatorname{Sq}}^{\mathrm{d}}\left( A\right) ,{\operatorname{Sq}}^{\varepsilon }\left( B\right) ,{\operatorname{Sq}}^{\varepsilon }\left( C\right) \) are collinear. | Proof. (a) The line joining \( {Y}_{\varepsilon } \) and \( {Z}_{\varepsilon } \) has equation \[ - {\varepsilon Sx} + {S}_{B}y + {S}_{C}z = 0 \] as is easily verified. This line clearly contains \( X = \left( {0 : - {S}_{C} : {S}_{B}}\right) \) . | Yes |
Corollary 5. Let \( X \) be a point defined with respect to the pedal triangle \( {A}_{P}{B}_{P}{C}_{P} \) triangle of \( P \) . The images of \( X \) after the pivoting as in Miquel’s theorem lie on a line. | Proof. Let \( {A}_{2}{B}_{2}{C}_{2} \) be the image of \( {A}_{P}{B}_{P}{C}_{P} \) after pivoting, and let \( Y \) be the image of \( X \) . Clearly triangles \( P{A}_{P}{A}_{2}, P{B}_{P}{B}_{2}, P{C}_{P}{C}_{2} \), and \( {PXY} \) are similar right triangles. This shows that \( Y \) lies on the line through \( X \) pe... | Yes |
Proposition 6. Let \( {A}_{\mathrm{a}}^{\varepsilon },{B}_{\mathrm{b}}^{\varepsilon } \) and \( {C}_{\mathrm{c}}^{\varepsilon } \) be the traces of a point \( P = \left( {u : v : w}\right) \) . (a) The three points \( {D}_{\mathrm{a}}^{\varepsilon },{D}_{\mathrm{b}}^{\varepsilon } \) and \( {D}_{\mathrm{c}}^{\varepsilo... | \[ 4{a}^{2}{b}^{2}{c}^{2}{uvw} + {S}^{2}\mathop{\sum }\limits_{\text{cyclic }}u\left( {\left( {2{S}_{A} + {S}_{B}}\right) {v}^{2} + \left( {2{S}_{A} + {S}_{B}}\right) {w}^{2}}\right) = {\varepsilon S}\left( {2{S}^{2}{uvw} + \mathop{\sum }\limits_{\text{cyclic }}u\left( {\left( {2{c}^{2}{a}^{2} - {S}_{AB}}\right) {v}^{2... | Yes |
Theorem 1. Given \( a,{\ell }_{1},{\ell }_{2} > 0 \), there is a unique triangle \( {ABC} \) with \( {BC} = a \) , and the lengths of the bisectors of angles \( B, C \) equal to \( {\ell }_{1} \) and \( {\ell }_{2} \) if and only if\n\n\[ \sqrt{{\ell }_{1}^{2} + {\ell }_{2}^{2}} < {2a} < {\ell }_{1} + {\ell }_{2} + \sq... | ## 2. Uniqueness\n\nFirst we prove that if such a triangle exists, then it is unique.\n\nDenote the sidelengths of the triangle by \( a, x, y \) . If the angle bisectors on the sides \( x \) and \( y \) have lengths \( {\ell }_{1} \) and \( {\ell }_{2} \) respectively, then from (1) above,\n\n\[ y = \left( {a + x}\righ... | Yes |
Corollary 3 (Steiner-Lehmus theorem). If a triangle has two equal bisectors, then it is an isosceles triangle. | Indeed, if the bisectors of the angles \( A \) and \( C \) of triangle \( {ABC} \) are equal, then triangle \( {ABC} \) is congruent to \( {CBA} \), and so \( {AB} = {CB} \) . | Yes |
Theorem 3. Let \( \mathcal{L} \) be a line through the orthocenter of a triangle \( {ABC} \) . The reflections of \( \mathcal{L} \) in the sidelines of \( {ABC} \) are concurrent at a point on the circumcircle. | See [11, p.99] or [10, §333]. | No |
Theorem 3. Let \( \ell \) and \( {\ell }^{\prime } \) be two lines intersecting at \( P \), tangent to the same inscribed conic \( \mathcal{E} \), and \( \mathrm{d} \) be a line not passing through \( P \) . Let \( X, Y, Z \) (respectively \( \left. {{X}^{\prime },{Y}^{\prime },{Z}^{\prime };{X}_{\mathrm{d}},{Y}_{\math... | An equivalent condition is that \( A, B, C \) and the vertices of the triangle with sidelines \( \ell ,{\ell }^{\prime } \), d lie on a same conic. More generally, consider points \( {X}_{\mathrm{d}}^{\prime },{Y}_{\mathrm{d}}^{\prime } \) and \( {Z}_{\mathrm{d}}^{\prime } \) such that the cross ratios\n\n\[ \left( {X,... | Yes |
Corollary 4. The midpoints of \( X{X}^{\prime }, Y{Y}^{\prime }, Z{Z}^{\prime } \) lie on a same line \( {\mathrm{d}}^{\prime } \) if and only if \( \ell \) and \( {\ell }^{\prime } \) touch the same inscribed parabola. In this case, if \( \ell \) and \( \ell \) touch the parabola at \( M \) and \( {M}^{\prime },{\math... | An equivalent condition is that the circumhyperbola through the infinite points of \( \ell \) and \( {\ell }^{\prime } \) passes through \( P \). | No |
Corollary 2. The two chains \( \left\{ {\cdots ,{\alpha }_{-2}^{ + },{\alpha }_{-1}^{ + },{\alpha }_{0}^{ + },{\alpha }_{1}^{ + },{\alpha }_{2}^{ + },\cdots }\right\} \) and \( \left\{ {\cdots ,{\alpha }_{-2}^{ - }}\right. \) , \( \left. {{\alpha }_{-1}^{ - },{\alpha }_{0}^{ - },{\alpha }_{1}^{ - },{\alpha }_{2}^{ - },... | From the inverted skewed arbelos (see Figure 5), it is easy to see that the circles \( {\alpha }_{p}^{ + },{\alpha }_{p}^{ - },{\beta }_{q}^{ + } \) and \( {\beta }_{q}^{ - } \) have two common tangent circles for any integers \( p \) and \( q \) . The line passing through the center \( {O}_{{\gamma }^{\prime }} \) of ... | Yes |
Theorem 5. Let \( n \) be an integer and \( a \neq b \) .\n\n(i) \( 1/{a}_{n}^{ + } = 1/{b}_{n}^{ + } \) if and only if the circle \( \gamma \) passes through \( {V}_{n \pm 1} \) or \( {W}_{n \pm 1}^{+ + } \) . If \( \gamma \) passes through \( {V}_{n \pm 1} \) , | \n\[ \frac{1}{{a}_{n}^{ + }} = \frac{1}{{b}_{n}^{ + }} = {\left( n\left( \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}\right) + \frac{1}{\sqrt{{r}_{\mathrm{A}}}}\right) }^{2} \]\n\n(1)\n\nand if \( \gamma \) passes through \( {W}_{n \pm 1}^{+ + } \) ,\n\n\[ \frac{1}{{a}_{n}^{ + }} = \frac{1}{{b}_{n}^{ + }} = {\left( \left( n... | Yes |
Corollary 6. If \( \gamma \) is the common external tangent of \( \alpha \) and \( \beta \), touching these circles from above, then (i) \( {a}_{1}^{ + } = {b}_{1}^{ + } \) ,(ii) \( {a}_{-1}^{ - } = {b}_{-1}^{ - } \) ,(iii) \( {a}_{-1}^{ + } = {b}_{1}^{ - } \) ,(iv) \( {a}_{1}^{ - } = {b}_{-1}^{ + } \), and\n\n\[ \text... | Proof. Since \( 1/\sqrt{a},1/\sqrt{b},1/\sqrt{{r}_{\mathrm{A}}} \) satisfy the triangle inequality, relation (v) immediately follows from Theorem 5. | No |
Theorem 7. Any circle touching \( \alpha \) and \( \beta \) at points different from \( O \) passes through \( {V}_{z \pm 1} \) for some real number \( z \) . The proper circle touching \( \alpha \) and \( \beta \) at points different from \( O \) and passing through \( {V}_{z \pm 1} \) for a real number \( z \neq \pm ... | Proof. We again invert the circles \( \alpha ,\beta \) and \( \gamma \) in the circle centered at \( O \) and with radius \( 2\sqrt{ab} \) as in the proofs of Theorems 1 and 5 and use the same notation. The circle \( \gamma \) is then carried into the circle \( \gamma \) with radius \( {c}^{\prime } = a + b \), because... | Yes |
Theorem 9. The circle inscribed in the curvilinear triangle formed by \( {\gamma }_{0} \), the \( y \) - axis, and one of the twin circles of Archimedes touching \( \beta \) is congruent to \( {\zeta }_{0,1}^{\alpha } \) . | To prove this theorem, we use the following result of the old Japanese geometry [7] (see Figure 12):\n\nLemma 10. Assume that t | No |
Lemma 10. Assume that the circle \( C \) with radius \( r \) is divided by a chord \( t \) into two arcs and let \( h \) be the distance from the midpoint of one of the arcs to \( t \) . If two externally touching circles \( {C}_{1} \) and \( {C}_{2} \) with radii \( {r}_{1} \) and \( {r}_{2} \) also touch the chord \(... | Proof. The centers of \( {C}_{1} \) and \( {C}_{2} \) can be on the opposite sides of the normal dropped on \( t \) from the center of \( C \) or on the same side of this normal. From the right triangles formed by the centers of \( C \) and \( {C}_{i}\left( {i = 1,2}\right) \), the line parallel to \( t \) through the ... | Yes |
Theorem 11. Let \( z \) and \( w \) be real numbers.\n\n(i) The circle \( {\epsilon }_{z} \) intersects \( \alpha \) and \( {\gamma }_{z} \) perpendicularly at their tangency point and the line segment \( A{V}_{z} \) also passes through this point.\n\n(ii) Let \( w \neq 0 \) . The circle \( {\epsilon }_{z} \) is orthog... | Proof. We once again invert the circles in the circle centered at \( O \) and with radius \( 2\sqrt{ab} \) as in the proofs of Theorems 1,5 and 7 and use the same notation.\n\nThe circle \( {\gamma }_{z} \) is then carried into the circle \( {\gamma }_{z}{}^{\prime } \) touching \( {\alpha }^{\prime } \) at a point wit... | Yes |
Theorem 12. Let \( n \) be an arbitrary integer and \( a \neq b \) .\n\n(i) \( 1/{a}_{n}^{ + } = 1/{b}_{-n}^{ + } \) if and only if the circle \( \gamma \) passes through \( {X}_{n, \pm } \) or \( {Z}_{n, \pm }^{+ + } \) . If \( \gamma \) passes through \( {X}_{n, \pm } \) , | \n\[ \frac{1}{{a}_{n}^{ + }} = \frac{1}{{b}_{-n}^{ + }} = {\left( n\frac{\sqrt{a + b}}{\sqrt{a} - \sqrt{b}} - 1\right) }^{2}\frac{1}{{r}_{\mathrm{A}}} \]\n\nand if \( \gamma \) passes through \( {Z}_{n, \pm }^{+ + } \) ,\n\n\[ \frac{1}{{a}_{n}^{ + }} = \frac{1}{{b}_{-n}^{ + }} = {\left( \left( n\frac{\sqrt{a + b}}{\sqr... | Yes |
Lemma 13. Let \( {A}_{0}{B}_{0} \) be the diameter of the circle \( \gamma \) parallel to the \( x \) -axis and intersecting the \( y \) -axis at the point \( {O}^{\prime } \) . Let \( {a}_{0} = \left| {{A}_{0}{O}^{\prime }}\right| \) and \( {b}_{0} = \left| {{B}_{0}{O}^{\prime }}\right| \), where \( {A}_{0} \) and \( ... | Proof. Assume that \( \gamma \) touches \( \alpha \) and \( \beta \) internally and \( a < b \) (see Figure 15). Let \( {O}_{\alpha },{O}_{\beta } \) and \( {O}_{\gamma } \) be the centers of \( \alpha ,\beta \) and \( \gamma \) and \( F \) the foot of the normal dropped from \( {O}_{\gamma } \) to the \( x \) -axis. B... | Yes |
Theorem 14. Let \( {AO} \) and \( {BO} \) be the diameters of the circles \( \alpha \) and \( \beta \) on the \( x \) - axis. Let \( P \) and \( Q \) be the intersections of the circle \( \gamma \) with the \( x \) -axis, choosing \( P \) and \( Q \) so that \( A, P, Q, B \) follow in this order on the \( x \) -axis, i... | Proof. We use the same notation as in Lemma 13 and its proof. Assume that \( \gamma \) touches \( \alpha \) and \( \beta \) internally and \( a < b \) . According to Lemma 13, there is a real number \( k \), such that \( a = {a}_{0}k \) and \( b = {b}_{0}k \) . Hence,\n\n\[{\left| {O}_{\gamma }F\right| }^{2} = {\left( ... | Yes |
Proposition 1. The apices of the isosceles triangles on \( B{A}_{1} \) and \( {A}_{1}C \) are the points\n\n\[ \n{A}_{b} = \left( {-{a}^{2}w : 2{S}_{\varphi }v + \left( {{S}_{C} + {S}_{\varphi }}\right) w : \left( {{S}_{B} + {S}_{\varphi }}\right) w}\right) ,\n\]\n\n(1)\n\n\[ \n{A}_{c} = \left( {-{a}^{2}v : \left( {{S}... | Proof. Let \( {A}_{\varphi } \) be the apex of the isosceles triangle with base \( {BC} \) and base angle \( \varphi \) . It is well known that the point \( {A}_{\varphi } \) has the coordinates \( \left( {-{a}^{2} : {S}_{C} + {S}_{\varphi } : {S}_{B} + {S}_{\varphi }}\right) \) . The line \( {A}_{b}{A}_{1} \) is paral... | Yes |
Proposition 2. The equation of the line \( {\mathcal{L}}_{a} \) is\n\n\[ \left( {{S}_{B}{v}^{2} + {S}_{C}{w}^{2} + {S}_{\varphi }{\left( v + w\right) }^{2}}\right) x + {a}^{2}{w}^{2}y + {a}^{2}{v}^{2}z = 0. \] | Proof. For \( \varphi \in \left( {-\frac{\pi }{2},\frac{\pi }{2}}\right) \smallsetminus \{ 0\} \), the equation of the line joining \( {A}_{b} \) and \( {A}_{c} \) is\n\n\[ \left| \begin{matrix} x & y & z \\ - {a}^{2}w & 2{S}_{\varphi }v + \left( {{S}_{C} + {S}_{\varphi }}\right) w & \left( {{S}_{B} + {S}_{\varphi }}\r... | Yes |
Theorem 3. The triangle bounded by the lines \( {\mathcal{L}}_{a},{\mathcal{L}}_{b},{\mathcal{L}}_{c} \) is perspective with \( {ABC} \) . Their axis of perspectivity is the trilinear polar of the barycentric square of the point \( P \) . | Proof. Let \( {A}_{0} = {BC} \cap {\mathcal{L}}_{a},{B}_{0} = {CA} \cap {\mathcal{L}}_{b} \), and \( {C}_{0} = {AB} \cap {\mathcal{L}}_{c} \) . In homogeneous barycentric coordinates, these are the points\n\n\[ \n{A}_{0} = \left( {0 : - {v}^{2} : {w}^{2}}\right) ,\;{B}_{0} = \left( {{u}^{2} : 0 : - {w}^{2}}\right) ,\;{... | Yes |
For any point \( P \) and any angle \( \varphi \), the perspector of the triangles \( {ABC} \) and \( {A}_{2}{B}_{2}{C}_{2} \) is the point | \[ {K}_{P}\left( \varphi \right) = \left( {\frac{{a}^{2}}{-{S}_{B}{v}^{2}\left( {{w}^{2} - {u}^{2}}\right) + {S}_{C}{w}^{2}\left( {{u}^{2} - {v}^{2}}\right) + {u}^{2}{\left( v + w\right) }^{2}{S}_{\varphi }} : \cdots : \cdots }\right) \] Strictly speaking, the perspector \( {K}_{P}\left( \varphi \right) \) is not defin... | Yes |
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