Ίσες ακτίνες

The given circles ${\omega }_1$ and ${\omega }_2$ lie inside an angle of vertex $O$, touching the arms. A ray drawn from point $O$ intersects circle ${\omega }_1$ at points $A_1$ and $B_1$, and circle ${\omega }_2$ at points $A_2$ and $B_2$, such that $O{A}_1<O{B}_1<O{A}_2<O{B}_2$. 

Circle ${\gamma }_1$ touches the circle ${\omega }_1$ on the inside, and also touches the tangents drawn to circle ${\omega }_2$ from point $A_1$. Similarly, circle ${\gamma }_2$ touches the circle ${\omega }_2$ on the inside, and also touches the tangents drawn to circle ${\omega }_1$ from point $B_2$. Prove that the radii of the circles ${\gamma }_1$ and ${\gamma }_2$ are equal.

KöMaL Problems in Mathematics

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