IT IS THE QUESTION OF CIRCLES MAM/SIR let P be any point on the circumcircle of triangle ABC and perpendiculars - Maths - Circles

Question

IT IS THE QUESTION OF CIRCLES MAM/SIR
let P be any point on the circumcircle of triangle ABC and
perpendiculars PL ,PM ,PN are drawn on the lines through line
segment BC CA and AB respectively shhow that the points L M N are
collinear this LMN line is called as Simpsons line

Koka Sri Lakshmi Divya Sai · Accepted Answer

$To show that L,M,N are collinear,we need to verify \frac{A N}{N B} = \frac{B L}{L C} = \frac{C M}{M A} = 1 Note that AN=PAcos ∠ P A N NB=PBcos ∠ P B N BL=PBcos ∠ P B L LC=PCcos ∠ P C L CM=PCcos ∠ P C M MA=PAcos ∠ P A M \frac{A N}{N B} = \frac{B L}{L C} = \frac{C M}{M A} = \frac{\cos ∠ P A N \times \cos ∠ P B L \times \cos ∠ P C M}{\cos ∠ P B N \times \cos ∠ P C L \times \cos ∠ P A M} Now also that ∠ P A N = ∠ P C L ∠ P B L = ∠ P A M ∠ P C M = ∠ P B N \Rightarrow \frac{A N}{N B} = \frac{B L}{L C} = \frac{C M}{M A} = 1 So,L,M,N are collinear$

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IT IS THE QUESTION OF CIRCLES MAM/SIR let P be any point on the circumcircle of triangle ABC and perpendiculars PL ,PM ,PN are drawn on the lines through line segment BC CA and AB respectively shhow that the points L M N are collinear this LMN line is called as Simpsons line

IT IS THE QUESTION OF CIRCLES MAM/SIR

let P be any point on the circumcircle of triangle ABC and perpendiculars PL ,PM ,PN are drawn on the lines through line segment BC CA and AB respectively shhow that the points L M N are collinear this LMN line is called as Simpsons line