62nd Ukrainian National Mathematical Olympiad

Mark the midpoints of the arcs $AB$, $AC$, $BC$ that do not contain other points, by the points $W_C$, $W_B$, $W_A$ respectively (fig. 13). It is clear that $K$ and $L$ lie on $W_A{W}_C$ and $W_A{W}_B$ respectively. Then. Fig. 13 $$\begin{array}{rl}\angle (KL,PI)& =\angle (AI,AP)=\angle (A{W}_A,AP)=\angle (W_A{W}_C,W_{C}P)=\\ & =\angle (K{W}_C,W_{C}P)=\angle (W_A{W}_B,W_{B}P)=\angle (L{W}_B,W_{B}P),\end{array}$$

Archimedes' Lemma. If a circle is inscribed in a segment of another circle bounded by the chord $BC$ and touches the arc at the point $A_1$ and the chords at the point $A_2$ then the line $A_1{A}_2$ is a bisector $\angle B{A}_{1}C$ (fig. 14). Fig. 14