Question 94
In the interior of an acute triangle $ABC$ that is not isosceles, there are three distinct points $A_1, B_1, C_1$ such that $AB_1 : CB_1 = AB : CB$ and $AC_1 : BC_1 = AC : BC$. Let $A_2$ be the point symmetric to $A_1$ with respect to the line $BC$, let $B_2$ be the point symmetric to $B_1$ with respect to the line $AC$, and let $C_2$ be the point symmetric to $C_1$ with respect to the line $AB$. Then, all of the following conditions are satisfied.
$\bullet$ The four points $A, A_2, B, C_2$ are concyclic.
$\bullet$ The four points $A, A_2, B_2, C$ are concyclic.
$\bullet$ The four points $B, B_2, C, C_2$ are concyclic.
$\bullet$ None of the three points $A_2, B_2, C_2$ lie on the circumcircle of the triangle $ABC$.
Prove that the triangle $A_1B_1C_1$ and the triangle $A_2B_2C_2$ are similar.
$\bullet$ The four points $A, A_2, B, C_2$ are concyclic.
$\bullet$ The four points $A, A_2, B_2, C$ are concyclic.
$\bullet$ The four points $B, B_2, C, C_2$ are concyclic.
$\bullet$ None of the three points $A_2, B_2, C_2$ lie on the circumcircle of the triangle $ABC$.
Prove that the triangle $A_1B_1C_1$ and the triangle $A_2B_2C_2$ are similar.