[ Auxiliary circle [ Angle between tangent and chord $\quad]$

From a point $M$, lying outside a circle, two tangents are drawn to this circle. The distances from a point $C$, lying on the circle, to the tangents are $a$ and $b$. Find the distance from point $C$ to the line $A B$, where $A$ and $B-$ are the points of tangency.
Let $P, Q, N$ be the feet of the perpendiculars dropped from point $C$ to the lines $M A, M B, A B$ respectively. Then points $P$ and $N$ lie on the circle with diameter $A C$, and points $N$ and $Q$ lie on the circle with diameter $B C$. Therefore, $\angle C P N = \angle C A B = \angle C B Q = \angle C N Q$ (we applied the theorem about the angle between a tangent and a chord). Similarly, $\angle C N P = \angle C Q N$. Thus, triangles $P C N$ and $N C Q$ are similar by two angles. Consequently, $C N : C Q = C H : C N$, and

$C N^{2} = C P \cdot C Q = a b$.

## Answer

$\sqrt{a b}$.