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For three points $a,b,c$ on a circle, what is the maximum value of $ab+ac$ when points $b$ and $c$ are fixed.

I believe that the maximum value of $ab+ac$ is when $a$ is in the middle of the bigger arc formed by $b$ and $c$..

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You are correct. This might work as an explanation of why you are correct

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If the blue circle with centre $O$ is the original, and $A$ is also on the perpendicular bisector of $BC$ then the red ellipse is the locus of all points on the plane which would have constant sum of distances to the foci $B$ and $C$ adding up to $AB+AC$.

The red ellipse contains the grey circle with centre $D$ (the midpoint of $BC$) which touches $A$ since the grey circle can be seen as the red ellipse scaled horizontally, and the grey circle contains the blue circle because $O$ lies between $A$ and $D$ so the blue circle can be seen as the grey circle scaled vertically.

So all other points on the blue circle would have sum of distances to $B$ and $C$ less than $AB+AC$.

Henry
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  • Awesome intution. It helped me to understand why always the middle of the arc works. Thank you –  Aug 05 '21 at 08:08
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Hint 1: You want to maximize $\sin{(b)}+\sin{(c)}$ for a fixed $b+c$

Hint 2: $0<b+c<\pi$ and sine is convex in $(0,\pi)$

acat3
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  • You are trying to say sin(b)=sin(c). Would you please explain it? –  Aug 05 '21 at 08:10
  • @PMPNP if $f(x)$ is convex then $\frac{f(b)+f(c)}{2}\leq f(\frac{a+b}{2})$ and equality occurs at $b=c$ – acat3 Aug 05 '21 at 08:21
  • This means $\frac {sin(b)+sin(c)}{2} < sin(\frac{b+c}{2}) $ , and here b+c is constant so $sin(b)+sin(c) \leq 2 *constant$. How we can argue further. May be I am lost. Correct me if i am wrong –  Aug 05 '21 at 08:27
  • @PMPNP correct, so you have the maximum point when $b=c$. Recall that $AB+AC=2r(\sin{(B)}+\sin{(C)})$ – acat3 Aug 05 '21 at 08:30
  • I got the idea. By sine law, $AB+AC= 2r(sin (B)+ sin(C))$, and now to maximize $AB+AC$, we need to maximize $sin(B)+sin(C)$, then you apply the convexity thing considering the fact $\angle B +\angle C $ is fixed. Finally, I get $AB+AC \leq 2r * 2 *constant$. Why is it so when b=c, it gives maximum. Can you suggest some article to read about these kinds of stuff? –  Aug 05 '21 at 08:45