Central and Twin Tetrahedra

H. Katsuura

Journal for geometry and graphics, Tome 27 (2023) no. 2, pp. 119-126.

Résumé

\def\vv#1{\overrightarrow{#1}} Given a tetrahedron $T$, the tetrahedron $T'$ constructed by connecting the four centroids of its faces is called the \emph{central tetrahedron} of $T$. A tetrahedron $T$ can be inscribed in a parallelepiped $W$ so that the edges of $T$ are the diagonals of the faces of $W$. By drawing the remaining six diagonals on the faces of the parallelepiped $W$, we obtain a new tetrahedron $T^\star$, and call it the \emph{twin tetrahedron} of $T$. Let $S^\star$ and ${S^\star}'$ be the circumcenters of $T^\star$ and ${T^\star}'$, respectively. We will prove that all tetrahedra $T$, $T'$, $T^\star$, and ${T^\star}'$ have the centroid in common, say $P$, and the five points $S$, ${S^\star}'$, $P$, $S'$, and $S^\star$ are collinear in this order such that $\vv{S'S^\star} = 2\vv{PS'}$, $\vv{SP} = 3\vv{PS'}$, $\vv{SS'} = 2\vv{S'S^\star}$, and $\vv{SS^\star} = 3\vv{S'S^\star}$. Moreover, we prove that (1) $T'$ and ${T^\star}'$ are twins, and (2) if the tetrahedron $T$ is orthocentric, then $T$, $T'$, $T^\star$, ${T^\star}'$ are orthocentric with orthocenters $S^\star$, ${S^\star}'$, $S$, and $S'$, respectively.

Détail

Classification : 51M04
Mots-clés : Central tetrahedron, twin tetrahedron, centroid, circumcenter, orthocentric tetrahedron, orthocenter