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.