Show that the largest triangle of given perimeter is equilateral.

To show that the largest triangle of a given perimeter is equilateral, we need to prove two things:

1. An equilateral triangle is indeed a triangle with the largest area for a given perimeter.
2. Any triangle with a larger area would necessarily have a larger perimeter.

Let's tackle these two points one by one:

1. Proving that an equilateral triangle has the largest area for a given perimeter:
We start by considering a general triangle with fixed perimeter. Let's assume its sides are labeled as a, b, and c. By the triangle inequality theorem, we have the following inequalities:
a + b > c,
a + c > b,
b + c > a.

Now, let's consider an equilateral triangle with side length s. Since all sides of an equilateral triangle are equal, we can also write the triangle inequality theorem as:
s + s > c,
s + c > s,
s + c > s.

From the above inequalities, we can conclude that 2s > c, which gives us the maximum value of c for a given s. Similarly, we can deduce that 2s > a and 2s > b, providing the maximum values of a and b. Thus, for a given perimeter, an equilateral triangle has the largest area.

2. Showing that any triangle with a larger area would have a larger perimeter:
Consider a general triangle with sides a, b, and c, and let A be its area. Now, imagine we have another triangle with sides x, y, and z, and its area is greater than A.

Using the triangle inequality theorem, we have:
a + b > c,
x + y > z.

By rearranging these inequalities, we get:
a > c - b,
x > z - y.

Now, let's compare the perimeters of the two triangles:
Perimeter of the first triangle = a + b + c
Perimeter of the second triangle = x + y + z

Since the second triangle has a larger area, we know that x + y + z > a + b + c. By substituting the inequalities we found earlier, we have:
(z - y) + (c - b) + c > a + b + c.

After simplifying, we get:
2c > 2a.

This shows that for any triangle with a larger area, its perimeter will also be larger.

Combining both points, we conclude that the largest triangle, for a given perimeter, is indeed equilateral.