

Proof of Heron's (Hero's) formula: Triangle
Name: Unknown
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
I am trying to find a complete proof of a formula that is either
Heron's formula or Hero's formula (I have seen it named both ways). The
formula computes the area of a triangle when only the lengths of the sides are
known. The formula: If a, b, and c are sides of a triangle and s =
(1/2)*(a+b+c) (s is half of the perimeter) then the area of the triangle is
the square root of s * (sa) * (sb) * (sc). The texts that have this
formula which I have been using either offer it with no proof, or say that the
proof is too long to be printed in that text. Can you refer me to a book that
has this proof, or find it?
Replies:
Given a triangle with sides a,b,c, semiperimeter s, and area A,
show that A^2 = s(sa)(sb)(sc). Solution: Drop an altitude (of length h) to
the side of length c. Then A = (1/2)ch, so A^2 = c^2 h^2 / 4. Use the
Pythagorean Theorem to obtain the following system:
(1) x^2 + h^2 = a^2
(2) y^2 + h^2 = b^2
(3) x + y = c
Substitute y = c  x into (2) and simplify. Then subtract
the result from (1). You will find that
2cx = a^2  b^2 + c^2.
From (1),
4c^2 h^2 = 4a^2 c^2  4c^2 x^2
= (2ac + 2cx) (2ac  2cx)
= (2ac + a^2  b^2 + c^2)(2ac  a^2 + b^2 c^2)
From (1),
4c^2 h^2 = 4a^2 c^2  4c^2 x^2
= (2ac + 2cx) (2ac  2cx)
= (2ac + a^2  b^2 + c^2)(2ac  a^2 + b^2 c^2)
= ((a+c)^2  b^2) (b^2  (ac)^2)
= (a+c+b)(a+cb)(b+ac)(ba+c)
= (2s)(2s2b)(2s2c)(2s2a)
= 16s(sa)(sb)(sc)
Chris Baker
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Update: June 2012

