geometry problem

[gray ravine]

Problems like this can be solved by looking at the extrema. The term $\frac{D_{A}}{V_{A}}+\frac{D_{B}}{V_{B}}+\frac{D_{C}}{V_{C}}$ goes towards $+\infty$ as $P$ gets close to $A$.

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Similarly, there has to be a minimum area. Obviously, it cannot be negative as all terms are positive. I will leave it in your hands to find that minimum value.

[shut patrol]

animeonfire

animeonfire

[gray ravine]

You don't need to define the actual function. Showing that one can be defined is only necessary to rectify solving by finding a minimum and a maximum value for the term.

The rest of the problem will automatically be solved by the "intermediate value theorem"

[shut patrol]

animeonfire

[gray ravine]

This is what you just said, yes

With minimum being "something positive"

By the way how fractions work, the term will be minimal if you find a P such that all V are maximal.

If you have the option, try plotting the triangle with different positions of P and see how the areas change

Can you find P such that at least one partial term D/V = 1?

And how does that affect the other terms?

If you're bored of trying random things, you can start looking at the mathematical relations between D and V.

In this case you might learn something using triangle similarities.