prove that diameter is the longest chord in the circle
Take any chord in a circle, say with endpoints AB. Let O be the center of the circle. Then segments AO and BO are radii of the circle. AOB is a triangle, and we know that the sum of the lengths of two sides of a triangle is always greater than or equal to the length of the third side. So:
|AB| ≤ |AO| + |BO|
(Here |AB| means length of AB.) Now the length of a diameter is twice the length of a radius, which is precisely equal to the quantity |AO| + |BO|.