Two circles $Ω_1$and $Ω_2$ with centers $Ο_1$ and $Ο_2$ respectively, intersect each other at points $C$ and $D$.
Two tangents $λ_1$ and $λ_2$ are drawn at point $C$, perpendicular to each other, cutting $Ω_1$ at point $A$ and $Ω_2$ at point $B$ respectively.
Points $Ο_1$and $Ο_2$ are joined, thus intersecting $Ω_1$ and $Ω_2$ at points $X$ and $Y$ respectively.
Furthermore, lines $AX$ and $BY$ are drawn which intersect each other at point $G$.
Find the measure of $∠GCA$
I could get the figure correctly but I could not go ahead.
Construct $AB.$ We know that $\frac{CO_{1}}{O_{1}A}=\frac{CO_{2}}{O_{2}B}=1.$
By converse of Thales Theorem, $AB||O_{1}O_{2}$
So, $∠AXO_1=∠XAB$ ........$(1)$
We know that $AO_1=O_{1}X$
So, $∠AXO_1=∠O_{1}AX$ ......$(2)$.
From $(1)$ and $(2),$ $AX$ is the angular bisector of $∠O_{1}AB.$
Similarly, $BY$ is the angular bisector of $∠O_{2}BA$
Hence, $G$ is the incenter of $ΔABC.$
Thus, $CG$ also bisects $∠ACB.$
Finally, $$∠GCA=\frac{90°}{2}=45°=\frac{π}{4}radians.$$ Ans.