Question 71
In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC]$, $A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the point of intersection of the tangents in $A$ and $C$ to the circumcircle of triangle $ABC$, $X$ be the point of intersection of lines $ZA$ and $A_1C_1$ and $Y$ be the point of intersection of lines $ZC$ and $A_1C_1$.
a) Prove that $T$ is the incircle of triangle $XYZ$.
b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$.
c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.
a) Prove that $T$ is the incircle of triangle $XYZ$.
b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$.
c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.