Question

LMNO is a parallelogram. A circle through L and M is drawn such that it intersects MN at A and LO at B. Prove that ABON is a cyclic quadrilateral.​

Answer 1

Question

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Answer

Given ABCD is a parallelogram. A circle whose centre O passes through A, B is so drawn that it intersect AD at P and BC at Q To prove Points P, Q, C and D are con-cyclic.

Construction Join PQ

Proof ∠1 = ∠A [exterior angle property of cyclic quadrilateral]

But ∠A = ∠C [opposite angles of a parallelogram]

∴ ∠1 = ∠C ,..(i)

But ∠C+ ∠D = 180° [sum of cointerior angles on same side is 180°]

⇒ ∠1+ ∠D = 180° [from Eq. (i)]

Thus, the quadrilateral QCDP is cyclic.

So, the points P, Q, C and D are con-cyclic. Hence proved.

Answer 2

Answer:

ABCD is a Parallelogram A circle through A,B is

so drown that it is intersects AD at P, and BC

at Q prove the P,Q,C and D are concylic.

Solution :

In a quadrilateral ABDQ,

ABPQ is cyclic

∠DPQ=∠B

∠PQR=∠A ...(1)

We know, it the sum of adjacent angle is 180

∘

∠A+∠D=180

∘

∠B+∠C=180

∘

...(2)

From equation (1) and (2)

∠PQC+∠D=180

∘

∠DPQ+∠C=180

∘

...(3)

Sum of opposite angles in PQCD is 180

∘

∴P,Q,C and D is a concylic.