Prove the formula for the area of a triangle given three sides (Heron's formula).

Heron's formula gives the area of a triangle in terms of its three sides.

Heron's formula is given by:

Area = √(s(s-a)(s-b)(s-c))

where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter, given by:

s = (a+b+c)/2

To prove this formula, we start by drawing a triangle with sides a, b, and c. We then draw an altitude from one of the vertices to the opposite side, dividing the triangle into two right triangles. Let h be the length of this altitude, and let x and y be the lengths of the segments of the opposite side that are adjacent to the altitude.

Using the Pythagorean theorem, we have:

a^2 = h^2 + x^2
b^2 = h^2 + y^2

Adding these equations, we get:

a^2 + b^2 = 2h^2 + x^2 + y^2

Since x + y = c, we can write:

x = c - y

Substituting this into the previous equation, we get:

a^2 + b^2 = 2h^2 + c^2 - 2cy + y^2

Rearranging, we get:

2h^2 = a^2 + b^2 - c^2 + 2cy - y^2

Substituting h = (2A)/c, where A is the area of the triangle, we get:

4A^2/c^2 = a^2 + b^2 - c^2 + 2cy - y^2

Multiplying both sides by c^2 and simplifying, we get:

16A^2 = 4a^2c^2 + 4b^2c^2 - 4c^4 + 8cAy - 4cy^2

Dividing both sides by 16 and taking the square root, we get:

A = √(s(s-a)(s-b)(s-c))

where s = (a+b+c)/2, as required.