Question

In A ABC, P, Q and R are the midpoints of sides AB, AC and BC respectively.
Seg AS=side BC. Prove that : PQRS is cyclic.​

Answer 1

$\huge\boxed{\fcolorbox{cyan}{Yellow}{SOLUTION :-}}$

Given : In Δ ABC , P,Q ,R are the mid point of the sides BC , CA and AB respectively. Also AS⊥ BC.

To prove : PQRS is a cyclic quadrilateral.

Construction : Join SR , RQ, QP

Proof : In right angled Δ ASP , R is the mid point of AB

RB = RD

∠2= ∠1  ....(1)

Since , R and Q are the mid point of ABand AC , then

RQ ║ BC

RQ ║BP

Since OP ║RB , then quadrilatera BPQR is a parallelogram

∠1 = ∠3  ....(2)

∠2 = ∠3

∠2 + ∠4 = 180°   ( linear pair )

∠3 + ∠4 = 180 °  ( ∵ ∠2 = ∠3 )

Hence , quadrilateral PQRS is cyclic quadrilateral