id stringdate 2695-01-01 00:00:00 3788-01-01 00:00:00 | question stringlengths 132 888 | auxiliary_line_description stringlengths 4 409 | answer stringlengths 1 52 | original_image imagewidth (px) 130 3.91k | auxiliary_line_image imagewidth (px) 32 1.6k |
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3787 | As shown in the figure, in the cube ABCD–A₁B₁C₁D₁, let M be the midpoint of segment AB, with AB = 2, and let N be the midpoint of segment B₁C₁. Point P is a moving point on segment D₁N. A plane α passes through segment MC and is perpendicular to segment DP; it intersects DP at point E. Find the minimum volume of the triangular pyramid P–MCE. | null | (20-4√5)/15 | ||
3788 | As shown in the figure, consider the right triangular prism ABC–A₁B₁C₁, where CA is perpendicular to CB, and CA = CB = CC₁ = 2. Let F be the midpoint of segment CC₁, and let points D and E be moving points on edges AA₁ and BB₁, respectively. Let θ be the angle between planes DEF and ABC. When θ reaches its maximum, there exists a point M on the intersection line l of planes DEF and ABC. Compute the maximum value of the sine of the angle between line C₁M and plane ABC. | null | √6/3 |
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