# Are the face–centroid pyramids of a convex congruent-faced polyhedron congruent?

Let a convex polyhedron $P$ be given, all of whose faces are congruent. Consider any pyramid formed by a face of $P$ as its base and the centroid of $P$ as its vertex. Allowing congruence to admit reflection, are all such pyramids comprising $P$ congruent?

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Look at this. – dtldarek Feb 6 '13 at 23:29

To make this explicit: imagine one end cap consisting of the coordinates $(0, 0, 1+c)$, $(\pm 1, 0, c)$ and $(0, \pm1, c)$ and the other endcap (rotated 45 degrees) having the vertex coordinates $(0, 0, -(1+c))$ and $(\pm\frac{\sqrt2}{2}, \pm\frac{\sqrt2}{2}, -c)$. Then the squared distance between e.g. $(1,0,c)$ and $(\frac{\sqrt2}{2}, \frac{\sqrt2}{2}, -c)$ is $\left(1-\frac{\sqrt2}{2}\right)^2+\frac{1}{2}+4c^2$; setting this equal to 2 (i.e. the squared length of the other edges of the shape) gives $4c^2 = 1\!\frac{1}{2}-\left(1-\frac{\sqrt2}{2}\right)^2 = \sqrt{2}$, or $c=\dfrac{\sqrt[4]{2}}{2}$. By symmetry, all of the other intermediate edges of the central band are of the same length, so all of the triangles in the central band are equilateral triangles.