Monday, November 28th

Problem
Let $E$ be an ellipse and $E'$ be its reflection along one of its tangents. Find the locus of the foci of $E'$ as the tangent line to $E$ varies.

Solution
Let $A$ and $B$ be the foci of $E$. Then by considering the reflexions of $A$ and $B$ over the tangent lines to $E$ one can see that the locus of $A$ is homothetic to the original ellipse with center at $B$ and a factor of 2. Likewise the locus of $B$ is homothetic to $E$ with center at $A$ and a factor of 2.