6. A circle is inscribed in a trapezoid. Find the area of the trapezoid if the lengths of $a$ one of the bases and segments $b$ and $d$, into which one of the lateral sides is divided by the point of tangency (segment $b$ is adjacent to the given base) are known.
6. Let (Fig. 1) $A B C D$ be a trapezoid, $|D C|=a, 0$ be the center of the circle. $A D$ is a lateral side, which is divided by the point of tangency $K$ into segments $|A K|=d,|K D|=b$. In $\triangle A O D$, the angle $A O D$ is a right angle, and the height $O K$ is the radius of the circle, so $r=$ $=|O K|=\sqrt{\overline{b d}}$. Similarly, $\triangle C O B$ is also a right triangle; if $M$ is the point of tangency on side $C B$, then $|M B|=$ $=\frac{r^{2}}{|C M|}=\frac{b d}{a-b}$.

Answer: $S=\frac{a^{2}+a(d-b)}{a-b} \sqrt{b d}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-064.jpg?height=275&width=435&top_left_y=1150&top_left_x=165)

Fig. 1.

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-064.jpg?height=300&width=459&top_left_y=1149&top_left_x=612)

Fig. 2.