11266

Orthogonal Projections of the Edges of a Cube

This Demonstration shows that the sum of the squares of the lengths of the orthogonal projections of the edges of a cube with edge length to a plane equals .
Let the bottom corner of the cube (a trihedron) have bottom vertex and three sides , , of length . Let be perpendicular to with . Let the angles of to the three sides be , , . Take the trihedron as the axes of a coordinate system with . Then , , , and so [1, p.27].
The length of the projection of to is , and similarly for and . So the lengths of the projections of the three edges are , , . Now
.
The 12 edges of the cube are parallel in sets of four, so the sum of the squares of all the edge lengths equals .

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DETAILS

This problem was posed in [1, pp. 20, 27, 28].
Reference
[1] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.
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