Properties and characteristics of the tangent [

A circle, the center of which lies on the hypotenuse $AB$ of a right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$, respectively.

Find the angle $ABC$, if it is known that $AE=1, BD=3$.
Let $O$ be the center of the circle. Consider similar triangles $B D O$ and $O E A$.

## Solution

Let $O$ be the center of the circle, $x$ be its radius. Then $O D=O E=x, O D \perp B C$ and $O E \perp A C$. Right triangles $B D O$ and $O E A$ are similar, so

$B D: D O=O E: A E$, or $3: x=x: 1$, from which $x=\sqrt{3}$. Therefore, $\operatorname{tg} \angle B=D O / B D=\frac{\sqrt{3}}{3}$.

Thus, $\angle B=30^{\circ}$.

## Answer

$30^{\circ}$.