Question

Triangle ABC is inscribed in a circle. Point P lies on a circumscribed circle of a triangle be such that through point P. PN, PM and PL are perpendicular on sides of triangle (possibly by increasing sides). Prove that points N, M and L are collinear.​

Answer 1

ABPC is a cyclic quadrilateral

∠BPC = 180° - ∠A

AMPL is a cyclic quadrilateral

∠MPL = 180° - ∠A

Thus,  

∠BPC = ∠MPL

∠BPL + ∠LPC = ∠BPL + ∠BPM

Therefore, ∠LPC = ∠BPM -- 1

LCPN is a cyclic quadrilateral

∠LPC = ∠LNC -- 2

BMPN is a cyclic quadrilateral

∠BPM = ∠BNM -- 3

Thus,from equation 1 2 and 3

∠BNM = ∠LNC

Therefore, N M and I are collinear