D is the midpoint of side BC of triangle ABC; P and Q lie on sides BC and BA in such a way that triangle BPQ = ΔABC. Let’s prove that, DQ || PA.

Given.

D is the midpoint of side BC of triangle ABC; P and Q lie on sides BC and BA in such a way that triangle BPQ = ΔABC

Formula used.

If 2 triangles are on same base And having equal area then they lies between 2 parallel lines.

Median divides triangle in 2 equal parts

As D is mid-point of side BC

AD is median

Triangle ADC = triangle ADB = × triangle ABC

As we have given with

Triangle BPQ = triangle ABC

Triangle BPQ = triangle ADB

Triangle BPQ = triangle BQD + triangle DQP

Triangle ADB = triangle BQD + triangle DQA

By subtracting triangle BQD from both sides

triangle DQP = triangle DQA

As both triangles triangle DQP , triangle DQA are equal

And both lies on same base DQ

Hence both triangle ’s lies between parallel lines DQ and PA

∴ DQ || PA