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,