# Q : 10     In any triangle ABC, if the angle bisector of   and perpendicular bisector of BC                intersect, prove that they intersect on the circumcircle of the triangle ABC.

Given :In any triangle ABC, if the angle bisector of   and perpendicular bisector of BC  intersect.

To prove : D lies on perpendicular bisector BC.

Construction: Join BD and DC.

Proof :

Let ABD = 1 ,  ADC = 2 , DCB =3 , CBD = 4

1 and 3 lies in same segment.So,

1 = 3    ..........................1(angles in same segment)

similarly, 2 = 4  ......................2

also,       1=2 ..............3(given)

From 1,2,3 , we get

3 = 4

Hence,  BD = DC (angles opposite to equal sides are equal )

All points lying on perpendicular bisector BC will be equidistant from B and C.

Thus, point D also lies on perpendicular bisector BC.

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