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.
Triangle in a circle
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circles
1 Answers
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Preliminary note:
An inscribed triangle is isosceles iff the perpendicular bisector of its base goes through the center of the circle:
In our case:
We have an inscribed triangle $ABC$ and we consider the angle bisector $a$:
The angles $CAD$ and $DAB$ equal. As a result (inscribed angle theorem) $CD=DB$. So we have an isosceles triangle $CDB$. This is the triangle to whose base we construct the bisecting perpendicular $p$.
Because of the statement mentioned as preliminary note, $p$ will go through the origin and the point $D$ produced by $a$.