34. The radius of the circumcircle of an acute-angled triangle $A B C$ is 1. It is known that the center of the circle passing through the vertices $A, C$ and the orthocenter of triangle $A B C$ lies on this circumcircle. Find the length of side $A C$.
34. Let the angles $A, B$ and $C$ be denoted by $\alpha, \beta$ and $\gamma$. Let $H$ be the orthocenter, and $O$ be the center of the circle passing through $A$, $H$ and $C$. Then

$$
\begin{aligned}
& \widehat{H O C}=2 \widehat{H A C}=2\left(90^{\circ}-\gamma\right) \\
& \widehat{H O A}=2 \widehat{H C A}=2\left(90^{\circ}-\alpha\right)
\end{aligned}
$$

But $\widehat{A O C}=180^{\circ}-\beta$ (since $B A O C$ is cyclic), $2\left(90^{\circ}-\gamma\right)+$ $+2\left(90^{\circ}-\alpha\right)=180^{\circ}-\beta, \quad 360^{\circ}-2 \alpha-2 \gamma=180^{\circ}-\beta, \quad 2 \beta=180^{\circ}-\beta$, $\beta=60^{\circ},|A C|=2 R \sin \beta=2 \frac{\sqrt{\overline{3}}}{2}=V^{\overline{3}}$.