Q8 of 24 Page 157

In the triangle ABC, the midpoints of AB and AC are D and E respectively; the midpoints of CD and BD are P and Q respectively. Let us prove that, BE and PQ bisect each other.


In ΔBDC, as Q and P are the midpoints of BD and CD respectively.


So by applying the theorem:-


The line segment joining the midpoints of two side of a triangle is parallel to the third side and equal to half of it, we get,


and PQ || BC ……… (1)


Similarly, applying above theorem on ΔABC where D and E are midpoints of AB and AC respectively


and DE || BC ……… (2)


From equations (1) and (2),


⇒ PQ = DE and PQ||DE, ……… (3)


⇒ PQDE is a parallelogram.


Also in ΔBDE, Q is the midpoint to BD and QH||DE as PQDE is a parallelogram,


……… (4)


From equations (3) and (4), it proves that:



⇒ BE and PQ bisect each other


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