Math Is Fun Forum

Agnishom

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From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

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Heron's formula without Trig

I really think this is cool.

Construction:
ΔABC with BC = a; AC = b, AB = c is constructed and AD is drawn from A such that it intersects BC at D in right angles. Now, let us take BD = d & DC = a - d and AD = h

Proof:
In ΔABD, d[sup]2[/sup] + h[sup]2[/sup] = c[sup]2[/sup]
⇒ h[sup]2[/sup] = (c + d) (c - d) = c[sup]2[/sup] - d[sup]2[/sup]                                          .... (i)
In ΔACD, (a-d)[sup]2[/sup] + h[sup]2[/sup] = b[sup]2[/sup]
⇒ (a² - 2ad + d²) + (c² - d²) = b² [from (i)]

                                                                                            ....(ii)

Now, we know that the Area of Triangle(T) =

From (i) & (ii),

Substituting  ½(a+ b+ c) with s,

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

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Registered: 2010-06-20

Posts: 10,623

Re: Heron's formula without Trig

Arhhh! the complete version.  I hoped that was coming.

Even greater proof!   

Bob

ps.  You now seem to have this twice.  Now maybe it deserves to be here twice .... but,  ... I can delete the other thread if you wish. 

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: Heron's formula without Trig

Yes. Please delete the other thread. Actually, I had hit the Submit button instead of the Preview...

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

{7/3}

Member

Registered: 2013-02-11

Posts: 210

Re: Heron's formula without Trig

I know this proof,this proof is used in my text book.So can anyone post the proof using trig

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There are 10 kinds of people in the world,people who understand binary and people who don't.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: Heron's formula without Trig

Check out this too

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.