If a median of a triangle is also an angle bisector, is it also an altitude?

Question

If a median of a triangle is also an angle bisector, is it also an altitude? I need help with this

Answer 1

If a median is also an angle bisector, then  the triangle is an isosceles triangle. 

 

The proof is this statement is a bit tricky.   Call the base angles   Θ₁  and Θ₂ .     The law of sines can be used to show that   sin(Θ₁) =  sin(Θ₂).     There are two possibilities:    1)   Θ₁ =  Θ₂  or  2)  Θ₁ =  180 - Θ₂ .   However, 2) can be ruled out because it implies that Θ₁ +  Θ₂  =  180 which forces the vertex angle to be zero measure.   This leaves Θ1 = Θ2 , which means that the triangle is isosceles. 

 

The median divides an isosceles triangle into two congruent triangles by sss.   The two angles made by the intersection of the median and the base are equal measure by corresponding parts.  However their sum is 180 degrees.  Thus both of these angles must be right angles.  

 

Since the median meets the base at right angles, it is an altitude.