Solution:
Given: In the given figure, where there are pair of Quadrilateral, in which AP=A Q, BP=B Q, C P=C Q .
To Prove : A, B and C are collinear.
Construction: Join AC , the point where it intersects P Q is M.
Proof:
AP=A Q, C P=C Q,
So, the quadrilateral A P C Q is a kite.→→If in a quadrilateral One pair of adjacent sides are equal, then the quadrilateral is a kite.
As we know in a kite Diagonals bisect each other at right angles.
∠AMP=∠A M Q=∠CM P=∠C M Q=90°
Also, BP=B Q, C P=C Q.
So, the quadrilateral B P C Q is a kite.
∠B MP=∠B M Q=∠CM P=∠C M Q=90°
As you can see that , 1. ∠AMP +∠CM P=90°+90°=180°→→Shows Point A and C are in line.------------(1)
2. ∠B MP +∠CM P=90°+90°=180°→→→Shows Point B and C are in line.------------------(2)
Combining (1) and (2),
Shows that point A, B,C lie in a line.
It means Points A, B,C are Collinear.