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Consider a triangle $ABC$ as in the following figure: enter image description here

I have to find the max and min of the side $AC$, knowing that it is possible to move the point $A$ inside the circle of radius $R$. This is what I've done, but I don't know if it is correct and rigorous. Using the cosine theorem:

$AC^2 = AB^2 + BC^2 - 2\cdot AB\cdot BC\cdot cos(\beta)$

The max seems simple: the max is attained when all the three right-hand side terms are at their maximum, thus when $AB = R$ and $cos(\beta) = -1$.

For the min, given $AB$, the right-hand side of the cosine theorem is minimized when $cos(\beta) = 1$. To find $AB$, when $cos(\beta) = 1$ then it becomes:

$ AB^2 + BC^2 - 2\cdot AB\cdot BC = (AB - BC)^2$

which is minimized when $AB$ assumes the largest value possible, that is $AB = R$.

Is this a valid and rigorous proof? Are there more rigorous ways to prove it? Thanks

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    Do you *have* to do it using trig? This is very obvious geometrically. – 2017-02-24
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    @dxiv: I agree, it is obvious geometrically, but the exercise was to prove it let's say "formally". Is it possible to give a geometrical proof without using trig? – 2017-02-24
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    Yes, but too long for a comment so I posted it as an answer. – 2017-02-24

2 Answers 2

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Is it possible to give a geometrical proof without using trig?

Inasmuch as can be inferred from the "in the following figure" premise, point $C$ is outside the circle. This implies that line $CB$ will intersect the circle at two points $A_0,A_1\,$, the first one inside the segment $CB\,$, the other one outside it.

By the triangle inequality, $CA+AB \ge CB = CA_0 + A_0B = CA_0 + R$ for any point $A\,$, which is equivalent to $CA \ge CA_0 + R -AB\,$. For points $A$ inside the circle $AB \le R\,$, so it follows that $CA \ge CA_0 - R + R = CA_0$.

$CA \le CA_1$ follows entirely similarly.

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Your proof looks valid to me, the only thing I think is worth including is that the side $BC$ is constant, and that $AB$ and $\beta$ are independent variables - thus justifying your method of maximisation/minimisation.