[Area of a triangle (using semiperimeter and the radius of the inscribed or excircle). Radii of the inscribed, circumscribed, and excircles (other).

In a right triangle, the segments $a$ and $b$ are known, into which the point of tangency of the inscribed circle divides the hypotenuse. Find the area of this triangle.

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Let $S$ be the area of a right-angled triangle with legs $x$ and $y$, $r$ be the radius of the inscribed circle, and $p$ be the semiperimeter. Then $p=a+b+r$. Therefore,

$$
2 S=x y=(a+r)(b+r)=a b+(a+b+r) r=a b+p r=a b+S
$$

From this, we find that $S=a b$.

## Answer

$a b$.

