# Q : 10     In any triangle ABC, if the angle bisector of  $\small \angle A$ 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  $\small \angle A$ and perpendicular bisector of BC  intersect.

To prove : D lies on perpendicular bisector BC.

Construction: Join BD and DC.

Proof :

Let $\angle$ABD = $\angle$1 ,  $\angle$ADC = $\angle$2 , $\angle$DCB =$\angle$3 , $\angle$CBD = $\angle$4

$\angle$1 and $\angle$3 lies in same segment.So,

$\angle$1 = $\angle$3    ..........................1(angles in same segment)

similarly, $\angle$2 = $\angle$4  ......................2

also,       $\angle$1=$\angle$2 ..............3(given)

From 1,2,3 , we get

$\angle$3 = $\angle$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.

## Related Chapters

### Preparation Products

##### Knockout NEET Sept 2020

An exhaustive E-learning program for the complete preparation of NEET..

₹ 15999/- ₹ 6999/-
##### Rank Booster NEET 2020

This course will help student to be better prepared and study in the right direction for NEET..

₹ 9999/- ₹ 4999/-
##### Knockout JEE Main Sept 2020

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 12999/- ₹ 6999/-
##### Test Series NEET Sept 2020

Take chapter-wise, subject-wise and Complete syllabus mock tests and get in depth analysis of your test..

₹ 4999/- ₹ 2999/-