JMO 2025 Problem 5

Let $ABC$ be an acute-angled scalene triangle. Inside the triangle, distinct points $A_1, B_1, C_1$ are chosen such that $$\frac{AB_1}{CB_1} = \frac{AB}{CB} \text{and} \frac{AC_1}{BC_1} = \frac{AC}{BC}.$$ Let $A_2, B_2, C_2$ be the reflections of $A_1, B_1, C_1$ across lines $BC, AC, AB$ , respectively. These points satisfy the following conditions:

• The four points $A, A_2, B, C_2$ are concyclic.
• The four points $A, A_2, B_2, C$ are concyclic.
• The four points $B, B_2, C, C_2$ are concyclic.
• The three points $A_2, B_2, C_2$ do not lie on the circumcircle of $\triangle{ABC}$ .

Prove that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.