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An isosceles trapezoid with a side length of $l$ is circumscribed around a circle. One of the bases of the trapezoid is equal to $a$. Find the area of the trapezoid.

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Find the second base and the radius of the inscribed circle.

## Solution

The sum of the bases of the given trapezoid is $2 l$; the second base is $2 l$ - $a$; the segments into which the point of tangency of the inscribed circle divides the lateral side are $\frac{a}{2}$ and $\frac{2 l-a}{2}$; the radius of the inscribed circle is the geometric mean of these segments; the height of the trapezoid is equal to the diameter of the inscribed circle.

## Answer

$l \sqrt{a(2 l-a)}$