Question

D and E are respectively the points on equal sides AB and AC of an isoceles triangle ABC such that B,C,E and D are concyclic, as shown in the given figure, if O is the point of intersection of CD and BE, prove that AO is the bisector of the line segment DE.​

Answer 1

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$\huge\mathfrak\purple{Solution:-}$

⏩ As DECB is a concyclic quadrilateral and

angle EDB + angle ECB = 180° and angle DBC + angle DEC = 180° also

as angle EDB + angle ADE = 180° and angle AED + angle DEC = 180°

[LINEAR PAIRS],

Hence,

We can say that,

angle ADE = angle ACB

and

angle AED = angle ABC.

Now,

As the given triangle is isosceles, so we can say

angle ADE= angle ACB= angle AED = angle ABC.

So,

DE is parallel to BC and AD= AE and DB =EC and DECB is an isoceles trapezium.

So,

DC and EB will be equal and if they intersect at O, AO will be the median of the triangle ABC and triangle ADE as well.

and that implies that,

AO is the bisector of the line segment DE.

HENCE PROVED...!!!!...:-)

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