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

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Solution

R.E.F image
ABCD is a Parallelogram A circle through $A, B$ is
so drown that it is intersects $A D$ at $P,$ and $B C$
at $Q$ prove the $P, Q, C$ and $D$ are concylic.
Solution :
In a quadrilateral $A B D Q,$
$A B P Q$ is cyclic
$∠ D P Q = ∠ B$
$∠ P Q R = ∠ A$ ...(1)
We know, it the sum of adjacent angle is $180^{\circ}$
$∠ A + ∠ D = 180^{\circ}$
$∠ B + ∠ C = 180^{\circ}$ ...(2)
From equation (1) and (2)
$∠ P Q C + ∠ D = 180^{\circ}$
$∠ D P Q + ∠ C = 180^{\circ}$ ...(3)
Sum of opposite angles in PQCD is $180^{\circ}$
$∴ P, Q, C$ and $D$ is a concylic.

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