Points K are taken on side AB of square ABCD: AK = BK. Prove that triangle CDK is isosceles.

Consider a square ABCD and let K be a point on side AB such that AK = BK.

Consider triangles KBC and KAD. Since the corners of the square A and B are right, both triangles are right-angled. Also note AK = BK and BC = AD, since they are the sides of the square.

Then the triangles KBC and KAD, as right-angled triangles, are congruent in two legs.

Therefore, KC = KD. And from this it follows that the triangle CDK is isosceles.