Please prove?

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In the figure, D,E and F are the mid points of sides AB, AC and BC respectively and AG is perpendicular on BC. Prove that DEFG is a cyclic quadrilateral.

1 Answer
Apr 15, 2018

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Given:

In Delta ABC

D,E,F are midpoints of AB,ACand BC respectively and AG_|_BC.

Rtp:

DEFG is a cyclic quadrilateral.

Proof:

As D,E,F are midpoints of AB,ACand BC respectively,
By midpoints theorem of a triangle we have

DE"||"BC orGF and DE=1/2BC

Similarly

EF"||"AB and EF=1/2AB

Now in Delta AGB,angle AGB=90^@ Since AG_|_BC given.

So angle AGB=90^@ will be semicircular angle of the circle drawn taking AB as diameter i,e centering D,

Hence AD=BD=DG=>DG=1/2AB

So in quadrilateral DEFG

DG=EF and DE"||"GF"

This means the quadrilateral DEFG is an isosceles trapezium which must be cyclic one,