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wintersolstice

Real Member

Registered: 2009-06-06

Posts: 128

Proof of Heron's Formula

If you spot any mistakes please let me know so I can edit, though I'm hoping there won't be any

where

This is the formula for the area of a triangle whose sides are a,b and c

Proof:

(NOTE: this proof uses Pythagoras' Theorem so in (dia 3) there's a simple proof of that)

(here are the diagrams)

start with the following (dia 1)

now

multiplying by b

substituting that into the area formula

squaring both sides

according to (dia 2)

and according to the cosine rule

whose proof is as follows:

again (dia 1)

(left triangle)

and

(right triangle)

rearranging the first equation

substituting into the second

expanding the bracket

canceling the X squared and a little rearrangement

from the left triangle

which means

substituting

rearranging

squaring

now the formula derived from (dia 2)

rearranging

which means

multiplying both sides by

now

so

altering the first fraction on the right

so

expanding the bracket on the right (stage 1)

expanding the bracket on the right (stage 2)

simplifying and rearranging

multiplying by 16

adding some terms to the right (that all cancel)

rearranging

factorisation (stage 1)

factorisation (stage 2)

bit of splitting joining and adding terms

rearranging

factorisation (stage 3)

factorisation (stage 4)

bit more splitting joining and adding terms

rearranging

factorisation (stage 5)

factorisation (stage 6)

reordering

little more splitting

dividing by 16

which means

substituting

square rooting both sides

Last edited by wintersolstice (2016-02-16 08:22:19)

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Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Proof of Heron's Formula

It looks far more complicated than it is (or should be) but it's correct (apart from a negligible typo when you missed out a 2 and wrote

instead).

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mathaholic

Member

From: Earth

Registered: 2012-11-29

Posts: 3,251

Re: Proof of Heron's Formula

I agree with Nehushtan. So long and complicated

(I remember my math tutor when he made very long formulae)

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