Question

prove that in a triangle other than an equilateral triangle the angle opposite to the longest side is greater than two thirds of a right angle

Answer 1

Consider a triangle that is ALMOST equilateral. One angle must therefore exceed 60o and it is well known that the larger the angle, the larger is the side opposite, and the converse is also true. Therefore the angle opposite the longest side must exceed 60o (which is, of course, therefore > two thirds of a 90o angle.