Two tangent lines of a circunference intersect at the center of another one if are perpendicular

Question

Suppose $\gamma$ and $\alpha$ are Euclidean circles that are perpendicular and intersect at the points $P$ and $Q$. Prove that the two tangent lines to $\alpha$ at $P$ and $Q$ intersect at the center of $\gamma$. Conclude that the center of $\gamma$ must lie outside $\alpha$.

Answer 1

I will assume that by "perpendicular circles" you mean that the tangent to the intersection point are perpendicular. If it is so, then just observe that the segment connecting of the point $P$ to the center of the circle $\alpha$ is perpendicular to the tangent to $\alpha$ crossing at $P$ (by definition of tangent). Hence, the center of $\alpha$ lies on the tangent to $\gamma$. You can argue similarly for all the other cases.

Since no point of the tangent to $\alpha$ can lie inside $\alpha$ (otherwise it would intersect $\alpha$ in two points), it follows that the center of $\gamma$ lies outside $\alpha$ (the other case being symmetrical).