Among all triangles inscribed in a given circle, the equilateral one has the largest area.

Proof

The proof depends on the following

Lemma

Among all triangles inscribed in a given circle, with a given base, the isosceles one has the largest area.

The assertion of the lemma is quite obvious:

Among all inscribed triangles with a given base, the tallest one is isosceles and, therefore, it has the largest area, due to the standard formula A = b×h/2, where A, b, and h are the area, the base and the altitude of a triangle.

The lemma shows that for a triangle that has two unequal sides, there is another triangle (an isosceles one at that) with the same circumcircle but larger area. The only triangle for which no improvement is possible is equilateral.

The lemma also shows that in order to prove the statement we only need to look among isosceles triangles. Consider the diagram below:

Let α be the base angle of an isosceles triangle ABC. Then half the apex angle at C equals 90° - α. Let O be the circumcenter of the triangle and D and E the midpoints of sides BC and AB, respectively. Thus, in the diagram, we successively obtain

So to maximize the area of triangle ABC we need to find the maximum of function f(β) = sin(β)(cos(β) + 1), where β = 2α. The simplest way to do that is to compute and equate to 0 the derivative: