Isosceles triangle and scalene triangle

Question:

Given the base and vertical angle of a triangle show that its area is greatest when the triangle is isosceles.

My attempt:

For isosceles triangle (with base given 2x, and vertical angle let z):

Note: $$y$$ is got by $$(180 - z)/2$$

I calculated area to be: $$CD^2 * tan(y)$$ by using trigonometry and the fact that the altitude in isosceles triangle from vertex angle bisects the base.

I don't know how I can prove this area to be greatest.

• Do you have any background in calculus? It can easily be proved with derivatives. – Ari Dec 3 '14 at 16:13
• @Ari Sorry. I do not have experience with calculus. The book in which this question is given doesn't teach about calculus. I guess the question has to be solved without it. Thank you! – Gaurang Tandon Dec 3 '14 at 16:14
• Use Heron formula: $A=\sqrt{s(s-a)+(s-b)+(s-c)}$ and $2s=a+b+c$. – ulead86 Dec 3 '14 at 16:22
• Could you clarify which is the vertical angle? – ghosts_in_the_code Dec 3 '14 at 16:22
• @user45195 The one opposite the base i.e. $\angle BAC$ – Gaurang Tandon Dec 3 '14 at 16:25