The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.
Consider the cone with a vertex $D$ and touches the two spheres.The intersection curve of the cone and the base should be an ellipse which touches the edges of the base triangle and whose foci are isogonal conjugate to each other with respect to the base triangle.
bingbing666 wrote:
The intersection curve of the cone and the base should be an ellipse which touches the edges of the base triangle and whose foci are isogonal conjugate to each other with respect to the base triangle.
Wait why is this true? Why is $X$ and $Y$ the foci of the desired ellipse?
