geometry
If the median and bisector of one of its sides of a triangle coincide, then the height also coincides and the triangle is isosceles.
So, to see that is isosceles I used the idea of this question: If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.
Now, to see that the height also coincides, I tried to assume that it doesn't but I don't see any contradiction.
If 1 and 2 are the same, then the angle bisect or theorem tells us that the third sides are equal, hence the triangle is isosceles.
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ Proof without words :
Well, not quite. :-$)$ In order to grasp things intuitively, first fix two things: the size of the angle
$($in green$)$, and that of the angle bisector $($the segment connecting the two yellow dots$)$. Keeping
the foot of the bisector fixed $($lower yellow dot$)$, we let the other two vertices of the triangle slide
freely along the two lines which determine the given angle. Thus we have the two pairs of red
and blue points. The former determine an isosceles triangle, where all three lines $($median, bi–
sector, and height$)$ coincide. In the latter case, by moving one vertex higher, $($and thereby closer
to the upper tip of the triangle, determining the height to fall on the right half of the figure$)$, the
other one is inevitably lowered, causing the position of the median's foot to descend, by stretch–
ing the length of the left portion of the triangle's base $($uniting the lower-left blue dot to the foot
of the bisector$)$, while simultaneously shrinking its right side $($uniting the foot of the bisector to
the upper-right blue dot$)$.
Best Answer
The two triangles made by the median are congruent and the base is a straight line so each angle made by the base and the median is $180/2 =90$ which makes the median to be the height as well.