We consider four distinct points $A, B, C$ and $D$ in the canonical Euclidean space $\mathbb{R}^3$ such that $AB = BC = CD = DA = 1$, $AC = a > 0$ and $BD = b > 0$. We assume that the four distinct points $A, B, C$ and $D$ are not coplanar. We denote $I$ the midpoint of $[AC]$ and $J$ the midpoint of $[BD]$. a) Show that $(IJ)$ is the common perpendicular to the lines $(AC)$ and $(BD)$. b) By projecting the points $B$ and $D$ onto the plane containing $(AC)$ and perpendicular to $(IJ)$, show that $a^2 + b^2 < 4$.
We consider four distinct points $A, B, C$ and $D$ in the canonical Euclidean space $\mathbb{R}^3$ such that $AB = BC = CD = DA = 1$, $AC = a > 0$ and $BD = b > 0$. We assume that the four distinct points $A, B, C$ and $D$ are not coplanar. We denote $I$ the midpoint of $[AC]$ and $J$ the midpoint of $[BD]$.
a) Show that $(IJ)$ is the common perpendicular to the lines $(AC)$ and $(BD)$.
b) By projecting the points $B$ and $D$ onto the plane containing $(AC)$ and perpendicular to $(IJ)$, show that $a^2 + b^2 < 4$.