Question

If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

Answer 1

$\text{Let}△ABC\text{be a triangle.}$
$\text{The bisector of}\angle \text{A bisects BC}$
$\text{To prove:}△ABC\text{is isosceles}$
(i.e., AB = AC)
We know that the bisector of the vertical angle divides the base of the triangle in the ratio of the other two sides.
$\therefore \frac{AB}{AC}=\frac{BD}{BC}$
$\text{Thus}\frac{AB}{AC}=1(\therefore \text{given)}$
$\Rightarrow AB=AC$
Hence the Triangle is isosceles.