Given: ap=aq, bp=bq, cp=cq prove: a, b and c are collinear.

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.

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