yeah.... Heron did.

smary answer sk... Heron did... well i have the paper of deriving it... but im too lazy to scan it or type it up here lol

Can someone derive Stewart's Theorem for me? i forgot how to... just for those who don't know what i mean its the MAN + DAD = BMB + CNC with the cevian and the triangle... sorry for being so nondescriptive.

to silver knight those proofs are mean anyone got some that does not use the law of cosines?

hold on til tomorrow drew i have the paper in my locker in my school. it has the derivation to the herons formula using no trig rule. so wait. patience is a virture

hmm ok.. sry for doing this so late. yesterday i was slightly ill.. so i dint have the strength to do it yesterday but today is fine.

There is a triangle, with sides a, b, c. The altitude h is made so that a is the base. Altitude h splits a into parts p and q. c is to the left of the altitude, and so it p. b is the right of the altitude, and so is q. Thats the triangle and all the sides needed to prove Herons formula

h= c^2 - p^2 = b^2 - q^2 (important equation needed to be used later)

- p^2 = b^2 - q^2 - c^2

q^2 - p^2 = b^2 - c^2 = (q + p)(q - p)

q + p = a (because p and q are the 2 parts that make up side a)

b^2 - c^2 = a(q - p)= aq - ap (now leave alone for just a second)

a^2 = a(p + q) = ap + aq

a^2 + b^2 - c^2 = 2aq (important equation needed to be used later)

(Area of Triangle) K= (1/2)(a)(h)

K^2 = (1/4)(a^2)(h^2)

16K^2 = (16)(1/4)(a^2)(h^2)

= 4(a^2)(h^2)

= 4(a^2)(b^2 - q^2) (this is from above equation)

= 4(a^2)(b^2) - 4(a^2)(q^2)

= (2ab + 2aq) (2ab - 2aq)

= (2ab + a^2 + b^2 - c^2) (2ab - a^2 - b^2 + c^2) (from above equation)

= (a^2 + 2ab + b^2 - c^2) (c^2 - a^2 + 2ab - b^2)

= ((a + b)^2 - c^2) (c^2 - (a - b)^2) (Now factor)

16K^2 = (a + b + c)(a + b - c)(c + a - b)(c - a + b)

K^2 = [(a + b + c)(a + b - c)(c + a - b)(c - a + b)]/16

K^2 = [(a + b + c)/2] [(a + b - c)/2] [(c + a - b)/2] [(c - a + b)/2]

Lets say s = [(a + b + c)/2] (semi-perimeter)

K^2 = (s)(s - c)(s - b)(s - a)

K = sqrt[s(s - c)(s - b)(s - a)]

There ya go. enjoy :)

could you make it any longer?