Question 510285
To construct a coordinate proof, you will have to fix points first. Without loss of generality suppose that for isosceles triangle ABC, A is at (0,0), B is at (2a,0), and vertex C is at (a,b) (so that AC = BC).


Let D and E be the midpoints of AC and BC respectively. We want to prove that BD = AE. We can easily find the coordinates of D and E using the fact that they're midpoints:


*[tex \LARGE D = (\frac{a}{2}, \frac{b}{2})]


*[tex \LARGE E = (\frac{3a}{2}, \frac{b}{2})]


Then the distance from B to D is


*[tex \LARGE \sqrt{(\frac{3a}{2})^2 + (\frac{b}{2})^2}] (using the distance formula/Pythagorean theorem)


Similarly, the distance from A to E is the same amount. Hence BD = AE, and we are done.