Find the point on the line
Find the perimeter of the triangle.
What is the inradius of the octahedron with sidelength a?
1) Determine the height of one of two constituent square pyramids by considering a right triangle, using the fact that the height of the equilateral triangle is
2) Now cut the (solid) octahedron along 4 of these latter heights, i.e. across two non-adjacent vertices and two midpoints of parallel sides of the ''square base''. (Figure 1)
Consider the resultant rhombic polygon, whose sidelength is
So the question becomes: Is there a still simpler proof?