20170524, 02:37  #1 
"Rashid Naimi"
Oct 2015
Remote to Here/There
11111100001_{2} Posts 
Area of a Triangle as a Function of the Lengths of its 3 Sides
Area of a Triangle as a Function of the Lengths of its 3 Sides.
If you already know the answer please keep quiet. If you don't, try to come up with the function before googling it. 
20170524, 05:02  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3·43·73 Posts 
All you need is a piece of paper, draw a triangle as below,
then write: b^2 = x^2 + y^2 c^2 = ... (fill the blank) Area = a*y / 2 Now: solve for y and get rid of x, and you will have area as a function of a, b, c. It is, you know, seventh grade level effort. You can be done in under 5 minutes. Is that even a puzzle? 
20170524, 05:38  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2,017 Posts 
That is not a general function. For different types of triangles the solutions will be different. For example for a right angle triangle the height can be equal to a side, so the algo will have to be adjusted.
A hint would be that the solution is independent of a base and a height( so you don't need to draw the triangle to specify the base and the height). The other hint is that the solution is 2k years old (but you shouldn't underestimate the difficulty.). Last fiddled with by a1call on 20170524 at 05:44 
20170524, 06:18  #4 
Romulan Interpreter
Jun 2011
Thailand
2^{5}×5×59 Posts 
We have read your "dotted line mystery" posts in the other forum (after Ernst gave us the link in another thread on this forum), and we know about your struggle to rediscover the Heron formula, which we knew for about 40 years by now... but what Serge said is correct, and it sets you on the right track. Just make all the calculus to the end, and THINK. Stop bullshitting around.

20170524, 10:59  #5  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:


20170524, 11:03  #6 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2,017 Posts 
The 2k years old general solution:
Heron's Formula https://www.mathsisfun.com/geometry/heronsformula.html Last fiddled with by a1call on 20170524 at 11:14 
20170524, 14:25  #7 
Aug 2006
3^{2}·5·7·19 Posts 

20170524, 14:53  #8  
Feb 2017
Nowhere
10660_{8} Posts 
Quote:
Some things can be discerned without knowing the answer. Assume there is an algebraic formula F(a, b, c). Then 1) F(a, b, c) = F(a, c, b) = F(b,c,a) = F(b, a, c) = F(c, a, b) = F(c, b, a) 2) If k is a positive constant, then F(k*a, k*b, k*c) = k^2 * F(a, b, c) 3) Assuming 0 < a <= b <= c, and c = a + b, then F(a, b, c) = 0. 4) If 0 < a <= b <= c, and c < a + b, then F(a, b, c) is not a positive real number. 

20170524, 17:13  #9 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2,017 Posts 
Interesting elaboration Dr Sardonicus. I particularly like number 2 and 4.
I have boasted about having an exceptionally good (long term) memory. Yet I do not recall ever being taught this formula in school (only base times 1/2 the height). So I thought perhaps I was not the only one and some others might be as impressed by the ancients' ingenuity as I was. Last fiddled with by a1call on 20170524 at 17:15 
20170524, 21:53  #10  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:


20170525, 00:11  #11  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3×43×73 Posts 
Two trivial notes:
Quote:
When you do what is spelled out for you in post #2, you will find that the obtained solution has a, b and c entered symmetrically. ... so you think people 2000 years ago couldn't do what is shown in post #2?? Wow! What amazing depth of ignorance! The opposite is true: people 2000 years ago were exactly the same as today. If anything, (average) people now only became stupider! (They don't have to work as hard as "ancients"  they have calculators, smart phones, microwaves, airplanes etc etc. so they really don't have to think anymore. They just press buttons.) 

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