Finding maximum area of triangle

Hi, I have the following problem.

"An isosceles triangle ABC, where AB = AC and BāC = 2(alpha) is inscribed in a circle of radius r. Show that the area of the triangle ABC is a maximum if alpha = pi/6."

To be honest I have no idea where to start. Also, I'm not sure if I'm in the right subforum, so move me if need be.

Re: Finding maximum area of triangle

Well, I know at least that I'm looking for dA/d(alpha), which is equal to dA/dr * dr/d(alpha)

dA/dr would be the derivative of the area respect to r, but I'm not sure what the formula for the area of this triangle is.. heheh..

dr/d(alpha) I -think- I can get with the cosine rule, from where I get the relationship arccos((a^2 -2r^2)/(-2r^2)) = alpha.

Derivating that I get dr/da =

So yup, any help would be appreciated.

Re: Finding maximum area of triangle

Wikipedia has the following formula for area: where is the measure of the other two angles.

Re: Finding maximum area of triangle

The following link might be useful

Isosceles Triangle Equations Formulas Calculator - Area Geometry

Has many useful formulas for various triangles. One of which is the radius of a circumscribed circle

Switching it around you get

You can plug those into the area formula and go from there.

Re: Finding maximum area of triangle

Quote:

Recall that in any triangle,

. "

" is the area of the trianlgle.

If k is one of the equal sides, .........[1]

https://lh6.googleusercontent.com/-a...640/figure.png

Also, .........[Cosine Formula in ]

.........[2]

Quote:

Substitute [2] into [1]:

Quote:

This is the relation between

and area of the triangle.

I hope you can easily proceed from here...