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# let P be a point on circumcircle of triangle ABC. perpendiculars PL,PM and PN are drawn on BC,CA and AB respectively. show that L,M,N are collinear

Saurabh Koranglekar
one year ago
Dear student

To Prove: N, M, and L are collinear.
Construction: Join PA and PC.Proof: ∠PMA +∠ANP = 90° + 90° = 180°Therefore, points A, M, P, N are concyclic....∠PMN =∠PAN ................... (i)
(angles in the same segment)Again ∠PMC =∠PLC = 90°Therefore, quadrilateral PMLC is cyclic....∠PML +∠PCL=180° ................... (ii)
Now, ∠PAB +∠PCB = 180°and ∠PAN +∠PAB = 180°...∠PAN =∠PCB =∠PCL ................... (iii)
From Equations (i) (ii) and (iii) we get,∠PML +∠PCL=∠PML +∠PAN=∠PML +∠PMN = 180°Hence, N, M, L are collinear. (NML line is called Simpson's line.

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