Problem 9.2. Let $A B C D$ be a parallelogram and let $\Varangle B A D$ be acute. Denote by $E$ and $F$ the feet of the perpendiculars from the vertex $C$ to the lines $A B$ and $A D$, respectively. A circle through $D$ and $F$ is tangent to the diagonal $A C$ at a point $Q$ and a circle through $B$ and $E$ is tangent to the segment $Q C$ at its midpoint $P$. Find the length of diagonal $A C$ if $A Q=1$.

Ivaylo Kortezov
9.2. Let $D H \perp A C(H \in A C)$. Then $\triangle A H D \sim \triangle A F C$ and $\triangle C H D \sim \triangle A E C$. Hence

$$
\begin{aligned}
A C^{2} & =A H \cdot A C+H C \cdot A C \\
& =A F \cdot A D+A E \cdot C D \\
& =A Q^{2}+A E \cdot A B=A Q^{2}+A P^{2}
\end{aligned}
$$

Setting $Q P=P C=x$, we get the equation

![](https://cdn.mathpix.com/cropped/2024_06_03_f8ccd71b4db367e7c2a9g-064.jpg?height=464&width=573&top_left_y=1608&top_left_x=1112)
$(1+2 x)^{2}=1+(1+x)^{2} \Longleftrightarrow 3 x^{2}+2 x-1=0$, which has a unique positive root $x=\frac{1}{3}$. Therefore $A C=1+2 x=\frac{5}{3}$.