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?
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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 $b$.
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: