[ Inscribed, circumscribed, and exscribed circles; their radii ] [ Two tangents drawn from one point

The radius of the circle inscribed in an isosceles right triangle is $r$, and half the perimeter is $p$. Find the hypotenuse.

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The segments of tangents drawn from a point to a circle are equal to each other.

## Solution

The vertex of the right angle of the given triangle, the center of the inscribed circle, and the points of tangency with the legs form the vertices of a square with a side length equal to $r$. The sum of the leg and half the hypotenuse is equal to $p$. By subtracting $r$ from this sum, we find the hypotenuse.

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-13.jpg?height=400&width=486&top_left_y=1358&top_left_x=792)

## Answer

$p-r$.