maximal triangle area

[BMAD]

What is the maximal area of a triangle if its sides a,b,c satisfying the inequality: 0<=a<=1<=b<=2<=c<=3?

[DejMar]

For a triangle in Euclidean space, the maximal area is equal to 1.

[DejMar]

An alternate answer can be given for a triangle in non-Euclidean space. On a curved plane the definition of a triangle can permit each of line segments of the triangle to be of the maximal lengths, i.e. a = 1, b = 2 and c = 3. Yet, it matters infinitismally that these lengths are used without placing bounds on the curvature of the space, as the surface area of the curved space can approach infinity. Simple stated, with non-Euclidean space and no other bounds than the lengths of the line segments the maximal area can be

∞.
I doubt, though, that this is the answer sought, and will assume the problem was meant to be bound in the Euclidean plane, so I shall stick with my answer in post #2.

[nuurhasan]

If the maximul area is 1 for any triangle, then is this equation satisfied: 0<=a<=1<=b<=2<=c<=3? How?

[DejMar]

If the maximul area is 1 for any triangle, then is this equation satisfied: 0<=a<=1<=b<=2<=c<=3? How?

There was no claim that the maximal area is for ANY triangle, but a triangle that satisfies the requirements, i.e., 0<=a<=1<=b<=2<=c<=3, and the inferred restriction that it is in the Euclidean plane, the maximal area is 1. No Euclidean triangle exists where a=1, b=2, and c=3, thus the length of at least 1 side is smaller. By iterating through various values using Heron's formula you can see that the maximal area approaches 1

.

Edited November 18, 2014 by DejMar

[DejMar]

The sides of a triangle with the maximal area of 1 occurs at

a=1, b=2, c≈2.23606797.