The median of a triangle drawn from the apex bisects the angle from which it is drawn (and of course the base). Is the triangle isosceles?

Question

This converse intrigues me.

Drawing it, clearly shows the the median divides the triangle in two triangles which are NOT congruent in the basic sense.  (A,S,S). Can you please help me.

Answer 1

use the Angle Bisector Theorem

draw triangle ABC

let A be the top vertex

bisect angle A with a line segment drawn to segment BC at point D

if BC is bisected also, meaning that BD=DC, then the triangle is isosceles because of the Angle Bisector Theorem

this theorem states that the ratio of the length of BD to the length of DC is equal to the length of side AB to the length of side AC, in other words...

BD/DC=AB/AC

because BD=DC, the ratio is equal to 1

if 1=AB/AC, then AB must equal AC in length

therefore, triangle ABC is an isosceles triangle  (it could be an equilateral triangle also)