SE-SKOLORNAS 1973 Problem 3

$ABC$ is a triangle with $\angle A = 90^\circ$ , $\angle B = 60^\circ$ . The points $A_1$ , $B_1$ , $C_1$ on $BC$ , $CA$ , $AB$ respectively are such that $A_1B_1C_1$ is equilateral and the perpendiculars (to $BC$ at $A_1$ , to $CA$ at $B_1$ and to $AB$ at $C_1$ ) meet at a point $P$ inside the triangle. Find the ratios $PA_1:PB_1:PC_1$ .