What is the largest area from this following triangle?

Question

Given coordinates A(2,0), B(0,2), and C is in line x+y= 5. We want to form the triangle ABC from these coordinates.

What is the largest possible area for this triangle?

My thinking about this problem is firstly we need to find a couple of coordinates such that the coordinates is formed the special triangle such that equilateral, isosceles or right isosceles triangle.

By using a distance concept, I've discovered that we cannot form the equilateral triangle from this problem. Instead of that, I can only get the form of isosceles triangle with area $$3*sqrt{17}$$, based on condition the coordinate of C is (2,5;2,5).

My question: Is my thinking true? Or there is another method to find the coordinate C such that maximizes the area of triangle?

Answer 1

So here is how the things sketch out

The base of your triangle will be the line between (2,0) and (0,2). And the height of the triangle will be the (perpendicular) distance be the red and blue lines. SO no matter what triangle you draw between the lines with the base between (2,0) and (0,2) you always get the same area.

Answer 2

I don't think you will ever get an equilateral triangle in this case.

The area will be symmetric about the point (2.5,2.5) and as we approach infinity, the area of the triangle approaches zero (could be seen easily by drawing a figure). This implies the area will be maximum at the point of symmetry.