Question

The 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.

This statement is true or false ?

• A. True
• B. False

Answer 1

The correct option is A True

Let angle bisector of $\angle A$ intersect circumcircle of $△ABC$ at $D$.

Join $DC$ and $DB$.

$\angle BCD=\angle BAD[\because \text{Angles in the same segment}]$

$\Rightarrow \angle BCD=\angle BAD=\frac{1}{2}\angle A.....\left(1\right)[\because \text{AD is bisector of}\angle A]$

Similarly,

$\angle DBC=\angle DAC=\frac{1}{2}\angle A.....\left(2\right)$

From ${eq}^n\left(1\right)\&\left(2\right)$, we have

$\angle DBC=\angle BCD$

$\Rightarrow BD=DC[\because \text{sides opposite to equal angles are equal}]$

$\Rightarrow D$ lies on the perpendicular bisector of $BC$.

Hence proved that angle bisector of $\angle A$ and perpendicular bisector of $BC$ intersect on the circumcircle of $△ABC$.

Hence the given statement is true.

Hence the correct answer is $\left(A\right)\text{True}$.