10.129. A circle is inscribed in an isosceles trapezoid. One of the lateral sides is divided by the point of tangency into segments of length \( m \) and \( n \). Determine the area of the trapezoid.
Solution.

Let $A K=m, K B=n$ (Fig. 10.117). Then $K B=B M=M C=n$, $A K=A N=N D=m$.

We find the height of the trapezoid:

$h=B H=\sqrt{A B^{2}-A H^{2}}=\sqrt{(m+n)^{2}-(m-n)^{2}}=2 \sqrt{m n}$.

Thus, $S=\frac{1}{2}(B C+A D) h=(m+n) h=2 \sqrt{m n}(m+n)$.

Answer: $\quad 2 \sqrt{m n}(m+n)$