Drop an altitude (of length h) to the side of length c.
Then A = (1/2)c h,
So,
A2 = c2 h2 / 4.
Use the Pythagorean Theorem to obtain the following system:
(1) x2 + h2 = a2
(2) y2 + h2 = b2
(3) x + y = c
Substitute y = c - x into (2) and simplify.Then subtract
the result from (1).
You will find that
2cx = a2 - b2 + c2.
From (1),
4c2 h2 = 4a2 c2 - 4c2 x2
= (2ac + 2cx) (2ac - 2cx)
= (2ac + a2 - b2 + c2)(2ac - a2 + b2 -c2)
From (1),
4c2 h2 = 4a2 c2 - 4c2 x2
= (2ac + 2cx) (2ac - 2cx)
= (2ac + a2 - b2 + c2)(2ac - a2 + b2 -c2)
= ((a+c)2 - b2) (b2 - (a-c)2)
= (a+c+b)(a+c-b)(b+a-c)(b-a+c)
= (2s)(2s-2b)(2s-2c)(2s-2a)
= 16s(s-a)(s-b)(s-c)
Then A = (1/2)c h,
So,
A2 = c2 h2 / 4.
Use the Pythagorean Theorem to obtain the following system:
(1) x2 + h2 = a2
(2) y2 + h2 = b2
(3) x + y = c
Substitute y = c - x into (2) and simplify.Then subtract
the result from (1).
You will find that
2cx = a2 - b2 + c2.
From (1),
4c2 h2 = 4a2 c2 - 4c2 x2
= (2ac + 2cx) (2ac - 2cx)
= (2ac + a2 - b2 + c2)(2ac - a2 + b2 -c2)
From (1),
4c2 h2 = 4a2 c2 - 4c2 x2
= (2ac + 2cx) (2ac - 2cx)
= (2ac + a2 - b2 + c2)(2ac - a2 + b2 -c2)
= ((a+c)2 - b2) (b2 - (a-c)2)
= (a+c+b)(a+c-b)(b+a-c)(b-a+c)
= (2s)(2s-2b)(2s-2c)(2s-2a)
= 16s(s-a)(s-b)(s-c)