prove-that-a-triangle-inscribed-in-a-circle-of-radius-r-having-maximum-area-is-an-equilateral-triangle-with-side-3-r-

Question Number 145246 by gsk2684 last updated on 03/Jul/21

prove that a triangle inscribed

in a circle of radius r having maximum

area is an equilateral triangle with

side \sqrt{3} r .

Answered by Olaf_Thorendsen last updated on 03/Jul/21

Necessarily, by symmetry a = b = c

{AB}^{2} = {OA}^{2} + {OB}^{2} - 2 OA . OB . \cos 120 °

a^{2} = r^{2} + r^{2} - 2 r . r (- \frac{1}{2})

a^{2} = 3 r^{2} \Rightarrow a = \sqrt{3} r

The triangle is equilateral and :

S = \frac{\sqrt{3}}{4} a^{2} = \frac{\sqrt{3}}{4} (3 r^{2}) = \frac{3 \sqrt{3}}{4} r^{2}

Commented by gsk2684 last updated on 03/Jul/21

thank you .

Will you please explain how to prove

to prove required triangle is

an equilateral triangle

Commented by ajfour last updated on 03/Jul/21

c i r c l e i s c i r c u l a r .

Commented by peter frank last updated on 04/Jul/21

t h a n k y o u . c a n y o u g i v e a

d i a g r a m