If two circles meet at a unique point, then the line through their centers passes through that point

Question

Let two circles meet at a unique point: M. Prove (or disprove) that the line joining their centers passes through M.

My attempt:

$T_1$ is tangent for $C_1$ and will meet $C_1$ at a single point $M$.

$T_2$ is tangent for $C_2$ and will meet $C_2$ at a single point $M$.

BUT:

Will $T_1$ meet $C_2$ at single point $M$, or two points? How can we prove it is a single point?

Similarly: Will $T_2$ meet $C_1$ at single point $M$, or two points?

Prove for any general two circles satisfying the property that they meet at a single point.

Answer 1

Hint: assume it's a different point and use the triangle inequality.

Answer 2

Hint: reflect $M$ in the line joining the centres to some $M'$. Is $M'$ on both circles?