Maximum area of a triangle inscribed in a circle with radius r.

Question

We want to find the maximum area of a triangle inscribed in a circle with radius r and with constant difference of two of its angles.

If $a, b, c$ are the angles of the triangle, if we set, wlog that $a>b$, we need to have:

$a-b = k$ (constant) and

$a+b+c=180$, so $a+b = 180-c$

I know that in general, without the restriction of the $2$ angles fixed difference, the largest triangle is the equilateral. Any assistance is much appreciated (by the way this is not homework or anything; just challenge between friends).

@OrShahar thank you but I prefer with basic geometry; I am not familiar with Lagrange multipliers. — Raahithya Vemulakonda, Dec 13 '20 at 10:15

score 1 · Accepted Answer · edited Nov 16 '22 at 03:38

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Hint: use this formula"

$S=\frac{abc}{4R}$

Now we want S be maximum for a constant R. We apply sine rule:

$\frac a{ \sin \alpha}=\frac b { \sin \beta}=\frac c{\sin \gamma}=2R$

$\alpha +\beta=180-\gamma$

$\alpha - \beta =k$

These two relations give:

$\alpha = \frac {\pi}2+\frac {k-\gamma}2$

$\beta = \frac {\pi}2 -\frac{k +\gamma}2$

Plug these in S you get:

$S=2R^2 \sin{\left( \frac {\pi}2+\frac {k-\gamma}2 \right)} \sin { \left(\frac {\pi}2 -\frac{k +\gamma}2 \right)} \sin (\gamma)$

Now take derivative of S ,$\gamma$ is the variable, set ir to 0 find $\gamma$ and put it in S and find the value of S. It can be seen (for example plotting S in wolfram) that for any k , S is maximum when $\gamma=90^o$, that is triangle must be right triangle to have maximum ares for any k.

edited Nov 16 '22 at 03:38

Mailbox

896

answered Dec 13 '20 at 10:16

sirous

10,751

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Referring to the angles $a,b$, one of the constraints is $a-b = k$ (constant). – user2661923 Dec 13 '20 at 10:23
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Sirous thanks for your answer but it can't be $a=b=c$ because we want $a-b$ to be constant (zero is a special case). – Raahithya Vemulakonda Dec 13 '20 at 10:29
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@RaahithyaVemulakonda, sorry I did not read the statement carefully. I will work and edit my answer. – sirous Dec 13 '20 at 10:54
@sirous how do you take derivative for $γ$ if $γ$ is inside the sine function? – Harry Gartner Dec 13 '20 at 11:23
I mean derivative of S as a function of variable $\gamma$. I edited my answer. – sirous Dec 13 '20 at 13:00
@sirous, did you get a result? I put it in Wolframalpha and didn't get any! – Raahithya Vemulakonda Dec 13 '20 at 13:07
The formula $S=2R^2\sin\alpha\sin\beta\sin\gamma$ is well-known, so you can start with it right away. – g.kov Dec 13 '20 at 13:36
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@RaahithyaVemulakonda, yes I did, see the edit. it must be right triangle for any k. Good experience for me.thanks for good question. – sirous Dec 13 '20 at 15:59

Maximum area of a triangle inscribed in a circle with radius r.

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