Question 39
Let $ABC$ be an isosceles triangle with apex $C$ and $\angle ACB\lt90^\circ$. Let $X\ne C$ lie on side $AC$, and let $Y\ne C$ lie on side $BC$. Let $D$ be a point such that $DX\parallel AB$ and $AC$ is the internal bisector of $\angle BAD$. Similarly, let $E$ be a point such that $EY\parallel AB$ and $BC$ is the internal bisector of $\angle EBA$. Suppose there is a point $T$ on side $AB$ satisfying $XT=XD$ and $YT=YE$. Express $AX+BY$ in terms of the side lengths of $ABC$.