# Minimizing (and maximizing) the area of triangles

How would one solve questions like this one here in general? I have gotten an answer for that question, but I don't understand what's the intuition behind it. Can smoeone clearly explain how to minimize and maximize the area of triangles (as asked in the linked question) and the intuition behind it?

-

## 2 Answers

Typically you will be in a situation where the area of your triangle will be determined by two variables. (In case of your referred question the numbers $a$ and $b$ where $A=(a,0)$ and $B=(0,b)$.) However in order to minimize the area, you would like to solve it as an extreme value problem, i.e. write the area as a function of a single variable, and then setting it's derivative to 0. So you need some kind of way to connect your two variables. In the case of the question you mention, the link is the fact that the line drawn through the points $A$ and $B$ is at an equal distance of given points $P$ and $Q$. Using the fact that these distances are equal, allows you to write $a$ as a function of $b$, and hence the area as a function of $b$.

-
This is what I wanted, thanks. – JohnPhteven Dec 7 '12 at 16:10

The question in this form does not make much sense. You always minimize or maximize something under some conditions.

So, for example, you can minimize or maximize the area of triangles where the three points lie on the unit circle.

What can go wrong:

1. It can happen that the area of the triangles are arbitrarily large, so that there is no maximal area, but in that case, we can just say so (that the area is arbitrarily large).

2. It can happen that the area of all triangles under a given condition is smaller than 2, can be chosen arbitrarily close to 2, but never 2. In that case, one calls 2 the supremum instead of the maximum. (The same thing can happen with the minimum.) Again, this can be explicitly stated and proved.

3. It can happen that the maximum exists, but there is no good explicit way of expressing it. In that case, there is not much you can do, not every number has a good explicit representation.

-
If you looked at me question you'd understand what I mean, I meant that if you have some data, but you can have an infinite amount of triangles with the data, however, there is a minimum and maximum of the area of the triangle. If you still don't understand, look at my question. – JohnPhteven Nov 27 '12 at 15:55
@ZafarS I have read the linked question before writing my answer. Have you read my answer? – Phira Nov 27 '12 at 16:01
Yes, but this is not a satisfying answer to me, because I still don't understand the answer I've gotten to my question, ergo, I don't understand how to minimize/maximize the area of a triangle when given certain information. – JohnPhteven Nov 27 '12 at 16:08
So, you can either try to improve your question, so that I understand why my answer is not satisfying to you and adapt it to your needs, or you can hope that someone else comes along who tries to guess what you want in spite of the fact that you will likely downvote for not reading your mind. (I.e. what "certain information" is) – Phira Nov 27 '12 at 16:16
And I have asked you repeatedly what "like the one" means. What are the derivatives of functions like $e^x$? – Phira Nov 27 '12 at 16:20