InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
that BPBQ.10. In any triangle ABC, if the angle bisector of ZAand perpendicular bisectorof BCintersect, prove that they intersect on the circumcircle of the triangle ABC. |
|
Answer» Suppose angle bisector of angle A meets circumcircle with centre O at P, and OP is joined to meet BC at D. ...(i) Angle BAP = Angle CAP (Because AP is angle bisector)Angle BOP = 2 Angle BAPAngle COP = 2 Angle CAPSo, Angle BOP = Angle COP Therefore in triangles BDO and CDO, BO = CO(Radii of the same circle)Angle BOP = Angle COP (Proven above)DO = DO (Common side So triangles BDO and CDO are congruent. (SAS criterion)So, BD = CD and angle BDO = angle CDO (CPCTE)But angles BDO and CDO form linear pair.So, angle BDO + angle CDO = 180 degreeSo, angle BDO = half of 180 = 90 degree With BD = CD and angle BDO = 90 degree,OD is perpendicular bisector of BCOROP is perpendicular bisector of CD ...(ii)By (i) and (ii),P lies on angle bisector of angle A and perpendicular bisector of BC and is a point on the circle. |
|