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?
 A: 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:


*

*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).

*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.

*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.
A: 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 $b$.
