Question

PQR is a triangle in which ∠Q = ∠R. If S and T are respectively the points on side PQ and PR such that SQ = TR, then prove that Q, S, T and R are concyclic.

Answer 1

Answer:

Given:In △ABCP,Q and R are the midpoints of the sides

BC,CA and AB respectively.

Also,AD⊥BC

In a right-angled triangle, ADP,R is the midpoint of AB

∴RB=RD

⇒∠2=∠1 ........(1)

since angles opposite to the equal sides are equal.

Since,R and Q are the mid-points of AB and AC, then RQ∥BC

or RQ∥BP (by mid-point theorem)

Since,QP∥RB then quadrilateral BPQR is a parallelogram,

⇒∠1=∠3 ........(2)

since angles opposite to the equal sides are equal.

From equations (1) and (2),

∠2=∠3

But ∠2+∠4=180

∘

(by linear pair axiom)

∠3+∠4=180

∘

(∵∠2=∠3)

Hence ,quadrilateral PQRD is a cyclic quadrilateral.

So,points P,Q,R and D are con-cyclic.

Hence the statement is true.