We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.

AB, AC are two equal chords of a circle. Prove that the line segment which bisects ∠BAC passes through the center of a circle. cool

 Nov 16, 2018

Let the angle bisector be AD, where D intersects the circle


Join DB  and DC 


And we have two triangles - BAD  and CAD




And angle BAD  = angle CAD

And AD is common


So...by SAS, triangle BAD is congruent to triangle CAD


But  angle DBA  = angle DCA...so....they intersect equal arcs

So....arc DBA  = arc DCA

But the sum of these arcs = 360°

So.....each angle intercepts an arc = 180°

But......the measures of these angles = 1/2 of their intercepted arcs = 90°


But...in a circle....an inscribed angle of 90° has its enpoints on a diameter

So...DA  must be a diameter

And a diameter always passes through the center of a circle

But DA is the angle bisector

So....DA passes through the center of the circle


cool cool cool

 Nov 16, 2018
edited by CPhill  Nov 16, 2018

7 Online Users