Thread: Triangle

Let be an acute triangle. Let and be inscribed rectangles in . Also, let be the area of a polygon . Does have a maximum? If so, what is it? and range over all rectangles and ranges over all triangles.

So they probably gave as an acute triangle for some reason. I am not sure why we need both and , since they are both rectangles. If you just let range over all rectangles, wouldn't that "cover" ? (e.g. we could instead consider )? Now is the area of all the triangles inside right?

Any ideas? Use any derivative tests?

Or you could also consider right?

Originally Posted by particlejohn

Let be an acute triangle. Let and be inscribed rectangles in . Also, let be the area of a polygon . Does have a maximum? If so, what is it? and range over all rectangles and ranges over all triangles.

So they probably gave as an acute triangle for some reason. I am not sure why we need both and , since they are both rectangles. If you just let range over all rectangles, wouldn't that "cover" ? (e.g. we could instead consider )? Now is the area of all the triangles inside right?

Any ideas? Use any derivative tests?

Reason why the two inscribed rectangles were considered may be because they don't overlap each other....which is why even only one inscribed rectangle could be considered for the purpose of the question.

Yes, the [A(R) +A(S)] / A(T) has a maximum.
The A(T) is constant.
The A(R) and/or A(S) are variable, depending on how they are positioned inside the triangle.
If the positions of A(R) +A(S) is maximized, then the [A(R) +A(S)] /A(T) is maximized also.

So essentially the problem can be simplified as follows: What is the largest rectangle you can inscribe in an acute triangle?

From this we know that a maximum exists.

Originally Posted by particlejohn

So essentially the problem can be simplified as follows: What is the largest rectangle you can inscribe in an acute triangle?

From this we know that a maximum exists.

Yes, if only one rectangle is considered.

If there are two rectangles, and they differ from each other, they will overlap each other. I was wrong in assuming they should not overlap each other. Otherwise one wiil not be inscribed in the triangle.