In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.
Circles
1
Posts
2
Users
0
Likes
382
Views
0
20/07/2021 5:01 pm
Topic starter
In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.
Answer
Add a comment
Add a comment
1 Answer
0
20/07/2021 5:03 pm
Consider this diagram
Here, join BE and CE.
Now, since AE is the bisector of ∠BAC
∠BAE = ∠CAE
∴ arc BE = arc EC
This implies, chord BE = chord EC
Now, consider triangles ΔBDE and ΔCDE,
DE = DE (It is the common side)
BD = CD (It is given in the question)
BE = CE (Already proved)
So, by SSS congruency, ΔBDE ΔCDE.
∴ ∠BDE = ∠CDE
We know, ∠BDE = ∠CDE = 180°
∠BDE = ∠CDE = 90°
∴ DE ⊥ BC (hence proved).
Add a comment
Add a comment
Forum Jump:
Related Topics
-
Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.
3 years ago
-
Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively.
3 years ago
-
AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.
3 years ago
-
ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE, = AD.
3 years ago
-
Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.
3 years ago
Forum Information
- 321 Forums
- 27.3 K Topics
- 53.8 K Posts
- 1 Online
- 12.4 K Members
Our newest member: Stripchat
Forum Icons:
Forum contains no unread posts
Forum contains unread posts
Topic Icons:
Not Replied
Replied
Active
Hot
Sticky
Unapproved
Solved
Private
Closed