KöMaL 2006 A389

Point $P$ lies in the interior of the acute triangle ABC . The circumcenters of triangles ABC , BCP , CAP and ABP are $O$, $A_1$, $B_1$ and $C_1$, respectively. Prove that $\frac{\mathrm{area}\,(\Delta A_1B_1O)}{\mathrm{area}\,(\Delta ABP)} = \frac{\mathrm{area}\,(\Delta B_1C_1O)}{\mathrm{area}\,(\Delta BCP)} = \frac{\mathrm{area}\,(\Delta C_1A_1O)}{\mathrm{area}\,(\Delta CAP)}.$