An isosceles triangle has 2 congruent sides and two congruent angles. The easiest way to prove that a triangle is isosceles using coordinate geometry is to use the sides.

Steps to Coordinate Proof

Given the coordinates of the triangle's vertices, to prove that a triangle is isosceles

plot the 3 points(optional)
use the distance formula to calculate the side length of each side of the triangle.
If any 2 sides have equal side lengths, then the triangle is isosceles.

Problem 1

The coordinates of triangle BCD are B(8,2), C(11,13) and D(2,6). Using coordinate geometry, prove that triangle BCD is an isosceles triangle.

Plot points

Calculate all 3 distances

Draw your conclusion

Triangle BCD is isosceles because 2 of the sides have equal side length $$ \sqrt{130} $$ (ie are congruent)

Problem 2

Triangle ABC has coordinate A(-2,3) , B (-5,-4) and C (2,-1). Using coordinate geometry, prove that triangle BCD is an isosceles triangle.

Plot points

Calculate all 3 distances

The distances

If you calculated side lengths AB and BC first, you could stop -- the triangle must be isosceles at this point since you found 2 sides that are congruent. (Granted, the triangle still could be equilateral)