Let $A B C$ be the angles of a triangle and $R$ the radius of the circle passing through the vertices. Connect the feet of the altitudes of the triangle and calculate the sides, angles, and the radius of the circumscribed circle of the resulting triangle (the pedal triangle).
The angles and sides of the pedal triangle are: $\alpha_{1}=180^{\circ}-2 \alpha, \beta_{1}=180^{\circ}-2 \beta, \gamma_{1}=180^{\circ}-2 \gamma ; a_{1}=a \cos \alpha, b_{1}=$ $b \cos \beta, c_{1}=c \cos \gamma$ (K.M.L.IV.45.I.). But $a=2 R \sin \alpha \mathrm{s}$ so $a_{1}=2 R \sin \alpha \cos \alpha=R \sin 2 \alpha$; similarly $b_{1}=R \sin 2 \beta$ and $c_{1}=R \sin 2 \gamma$. The radius of the circumcircle of the pedal triangle is

$$
r=\frac{c_{1}}{2 \sin \gamma_{1}}=\frac{R \sin 2 \gamma}{2 \sin 2 \gamma}=\frac{R}{2}
$$

Number of solutions: 33.