Answer to Question #7208 in Analytic Geometry for Jessica

Question

How do you prove that "The altitude to the base of an isosceles triangle is also a median"?

Answer 1

Let ABC be an isosceles triangle with base AC, so AB=BC.
Let also BD be the
altitude.
We should prove that BD is also a median, so AD=DC.

Consider
triangles ABD and CBD.
Then we have the following:

1) Since ABC is
isosceles, we have that angle A = angle C.

2) The angles BDA and BDC are
right and so they are equal

3) It follows from 1) and 2) that the third
angles ABD and CBD are equal as well.

3) AB=CB

5) the side BD is
common for both triangles.

Thus due to 3), 4) and 5) thriangles ABD and
CBD are equal, since they have two equal sides and equal angles
between
them.
In particular, AD=CD, and so BD is median.