Proving that two tangent circles, in the case when one is insrcibed into the other, share common tangent line

Question

The problem:

Given two circles with different radii, they intersect each other only at one point $M$. The smaller circle is inscribed into the bigger circle. Prove that a tangent line to one circle is also tangent to the other cicle.

The figure:

I could prove that if a tangent line is tangent to the bigger circle, then it's also tangent to the smaller one, but I'm confused with the opposite case. On the picture tangent line initially belongs to the circle with the center $O_{1}$.

I've tried to prove that $O_{1}O + O_{1}M = OM$, or that $O_{1}M$ is also perpendicular to the tangent line of the smaller circle by finding some contradiction, but none of that worked. How to prove it?

Please, consider that it's only 8th grade curriculum and I need to prove it within the topic of: "Central and inscribed angles".

Answer 1

I don't know what's within the scope of your curriculum, but here's the outline for an argument based proof:

1. Consider the first circle, a point $M$ on it, and a line $L$ tangent at $M$.
2. The second circle shares only the point $M$ with the first circle.
3. Since line $L$ contains $M$, it must intersect the second circle at least at $M$.
4. If line $L$ is not tangent to the second circle, then it must intersect as a chord. So there is a second point $M'$ shared by the line and the second circle, and $MM'$ is the chord formed.
5. The second circle goes from $M$ to $M'$ on both sides of $MM'$. On one of these sides, it must enter the interior of the first circle.
6. $M'$ is outside the first circle, since it lies on line $L$ which touches the first circle only at $M$. So one of the arcs from $M$ to $M'$ would have to go from inside to outside the first circle, crossing it at some point $M^*$.
7. This implies $M^*$ is another point common to the two circles. But the problem statement says the two circles intersect at only one point.
8. This forces $M^* = M$, which is only possible when $M = M'$ as well.
9. Hence, the line $L$ cannot intersect the second circle at another point distinct from $M$. Therefore, it is tangent to the second circle.