V OBM

Question

An equilateral triangle ABC has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side a is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.

Accepted Answer

The key observation is that the midpoints of the edges of a regular tetrahedron form the vertices of a regular octahedron. So we can place a tetrahedron (side a) with its base on a plane next to an octahedron (side a) with its base on the same plane and with a face of each coinciding. Now the pyramid is just half an octahedron, so evidently if we rotate it from its initial position so that its apex coincides with the apex O of a regular tetrahedron (whose base is ABC), then one face of the pyramid is parallel to the base of the tetrahedron. Then the result follows immediately.