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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

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

Grade:11

1 Answers

Saurabh Koranglekar
askIITians Faculty 10335 Points
4 years 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.

Regards

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