Question 40
Let $AB\Gamma$ be an acute angled triangle inscribed in the circle $c(O, R)$ (with $AB \lt A\Gamma \lt B\Gamma$) and let $\Delta, E, Z$ be the touching points of the incircle of the triangle with the sides $B\Gamma$, $A\Gamma$, $AB$, respectively. The circumcircle of the triangle $AEZ$ (say, $(c_1)$) intersects the circle $(c)$ at point $A'$. The circumcircle of the triangle $B\Delta Z$ (say, $(c_2)$) intersects the circle $(c)$ at point $B'$. The circumcircle of the triangle $\Gamma\Delta E$ (say, $(c_3)$) intersects the circle $(c)$ at point $\Gamma'$. Prove that:
(α) The quadrilateral $\Delta EA'B'$ is cyclic.
(β) The lines $\Delta A'$, $EB'$ and $Z\Gamma'$ are concurrent.
(α) The quadrilateral $\Delta EA'B'$ is cyclic.
(β) The lines $\Delta A'$, $EB'$ and $Z\Gamma'$ are concurrent.