--- In

mathforfun@yahoogroups.com, "Daniel" <dbeyene@...> wrote:

>

> For A + B + C = 360 degrees

>

> Prove that [sin(A) + sin(B) + sin(C)] is maximum when A = B = C

>

>

>

> I kind of see a pattern on excel in favor the proof but just

wondering if

> there is a conclusive derivation.

>

>

>

> Daniel

Except for a factor (1/2) this is the area of the triangle formed by

three points on a unit circle where A,B,C are the arc angles between

pairs of the three radii from the center to the three points (the

area of a triangle formed by two radii of unit length with angle x

between them is (1/2)sin(x)).

--> The problem is equivalent to proving that if three points on a

circle are chosen to maximize the area of their triangle, it must be

equilateral (A = B = C = 120).

Suppose P1,P2,P3 are three points on a circle forming a triangle of

maximal area and that two sides, say P1P3 and P1P2 are not equal.

Then the point T on the circle at the top of the arc of the chord

P2P3 is farther from the chord P2P3 than P1, so the triangle TP2P3

has the same base (P2P3) as P1P2P3 but a higher altitude --> its area

is bigger, contradicting the assumption.