Circles and equal segments

[ins-]

Triangle $ABC$ is inscribed in a circle $k(O)$. $t$ is the tangent to $k$ through the vertex $C$. From the vertices $A$ and $B$ are drawn perpendiculars $AA_1$ and $BB_1$ to $t$. Through $A_1$, $B_1$ and $O$ is drawn a circle $k_1$ intersecting $AA_1$ and $BB_1$ at the points $A_2$ and $B_2$ respectively. Prove that $A_1A_2=B_1B_2$.

[Dave]

$A_1A_2B_2B_1$ - прямоугольник, т.к. $A_1A_2 \perp A_1B_1$, $A_1B_1 \perp B_1B_2$ и $A_1A_2B_2B_1$ вписан в окружность.

[ins-]

I'm sorry for the stupid problem and loosing your time.