D, E, F are the midpoints of the sides AB, BC and CA respectively of triangle ABC. AE meets DF at O. P and Q are the midpoints of OB and OC respectively. Prove that DPQF is a parallelogram.

Asked by Nitin | 3rd Jul, 2017, 07:49: PM

Expert Answer:

begin mathsize 16px style In space triangle ABC comma
straight D space and space straight F space are space the space mid minus points space of space AB space and space AC space and space
DF vertical line vertical line BC space and space 2 DF equals BC space space..... left parenthesis By space Mid minus point space theorem right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space..... left parenthesis straight i right parenthesis
In space triangle BOC comma
straight P space and space straight Q space are space the space mid minus points space of space BO space and space CO space and space
PQ vertical line vertical line BC space and space 2 PQ equals BC space space..... left parenthesis By space Mid minus point space theorem right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space..... left parenthesis ii right parenthesis
In space triangle ABO comma
straight D space and space straight P space are space the space mid minus points space of space AB space and space BO space and space
DP vertical line vertical line AO space and space 2 DP equals AO space space..... left parenthesis By space Mid minus point space theorem right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space..... left parenthesis iii right parenthesis
In space triangle ACO comma
straight Q space and space straight F space are space the space mid minus points space of space CO space and space AC space and space
FQ vertical line vertical line AO space and space 2 FQ equals AO space space..... left parenthesis By space Mid minus point space theorem right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space..... left parenthesis iv right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space have space DF equals PQ space and space DF vertical line vertical line PQ.
From space left parenthesis iii right parenthesis space and space left parenthesis iv right parenthesis comma space we space have space DP equals FQ space and space DP vertical line vertical line FQ.
Since space opposite space sides space of space quadrilateral space DPQF space are space parallel comma space
DPQF thin space is space straight a space parallelogram. end style

Answered by Rebecca | 4th Jul, 2017, 05:05: AM