[ Diameter, main properties ]  [ Signs and properties of the tangent ]

A circle passes through the midpoints of the hypotenuse $A B$ and the leg $B C$ of a right triangle $A B C$ and is tangent to the leg $A C$. In what ratio does the point of tangency divide the leg $A C$.
Use the theorem about the diameter of a circle perpendicular to a chord and the theorem about the midline of a triangle.

## Solution

Let $M$ be the midpoint of the hypotenuse $AB$, $N$ be the midpoint of the leg $BC$, $K$ be the point of tangency of the given circle with the line $AC$, and $P$ be the midpoint of the midline $MN$ of triangle $ABC$. The perpendicular to $AC$ through point $K$ passes through the center of the circle and bisects the chord $MN$ perpendicular to it, i.e., it also passes through point $P$. Therefore,

$$
CK = NP = \frac{1}{2} MN = \frac{1}{2} \cdot \frac{1}{2} AC = \frac{1}{4} AC.
$$

Thus, $CK : AK = 1 : 3$.

## Answer

$1: 3$, measured from vertex $C$.