CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Question 5
Prove that angle bisector of any angle of a triangle and perpendicular bisector of the opposite side. If intersect they will intersect on the circumcircle of the triangle.

Open in App
Solution

Given ΔABC is inscribed in a circle.
The bisector of A and the perpendicular bisector of BC intersect at point Q.


To prove that A, B, Q and C are concyclic.
Construction : Join BQ and QC

Proof
We have assumed that Q lies outside the circle.
In ΔBMQ and ΔCMQ
BM = CM [QM is the perpendicular bisector of BC]
BMQ=CMQ [each 90]
MQ = MQ [Common side]
ΔBMQΔCMQ [by SAS congruence rule]
BQ = CQ [by CPCT] … (i)
Also, BAQ=CAQ [given] ..........(ii)
From Eqs. (i) and (ii) , we can say that Q lies on the circle
[equal chords of a circle subtend equal angles at the circumference]
Hence, A, B, Q and C are concyclic.


flag
Suggest Corrections
thumbs-up
2
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Circles and Their Chords - Theorem 3
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon