IOM 2021 Problem 2

Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ so that $P$ lies between $B$ and $Q$ . The rays $AP$ and $AQ$ divide the angle $BAC$ into three equal parts. It is known that the triangle $APQ$ is acute-angled. Denote by $B_1,P_1,Q_1,C_1$ the projections of points $B,P,Q,C$ onto the lines $AP,AQ,AP,AQ$ , respectively. Prove that lines $B_1P_1$ and $C_1Q_1$ meet on line $BC$ .