20140822, 21:59  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}×3^{2}×7 Posts 
A triangle problem
HI Math People,
This problem comes from Tarence Tao's book, Solving Mathematical Problems. It is problem 1.1. A triangle has its lengths in an arithmetic progression, with difference d. The area of the triangle is t. Find the lengths and angles of the triangle. Best of luck. Regards, Matt 
20140822, 22:41  #2 
"Ed Hall"
Dec 2009
Adirondack Mtns
3·1,399 Posts 
Sounds like a 345 right triangle...
d=1 lengths 3,4,5 angles 90, ~36.86, ~53.13 t=3*4/2 
20140823, 01:15  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{5}·3·101 Posts 
All you need is Heron's formula.
[ TEX ] b=\sqrt{2(d^2+\sqrt{d^4+4/3 t^2})}[/ TEX ] a=bd c=b+d A=acos((b^2+c^2a^2)/(2bc))=acos((b+4*d)/2/(b+d)) B=acos((a^2+c^2b^2)/(2ac))=acos((b^2+2*d^2)/2/(b^2d^2)) C=acos((a^2+b^2c^2)/(2ab))=acos((b4*d)/2/(bd)) Last fiddled with by Batalov on 20140823 at 01:24 Reason: forgot the denominator... 
20140825, 05:42  #4 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}·3^{2}·7 Posts 
So I think the answer involves an expression in t.
Regards, Matt 
20140825, 06:19  #5 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10010111100000_{2} Posts 

20140825, 06:57  #6 
"Tapio Rajala"
Feb 2010
Finland
3^{2}×5×7 Posts 
I don't think he thinks that. (at least his comment does not imply it.)
EdH assumed the sidelengths to be integer and did not take t as an input. I understood the comment to be on this. For even with integer lengths the solution is not unique. Batalov's solution seems correct (or at least looks like a correct one, I did not do the math). 
20140825, 21:53  #7 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}×3^{2}×7 Posts 
ok, to post the solution from the book
defining the side lengths as bd, b, and b+d then, applying Heron's formula gives  t^2 = 3b/2 * (3b/2  b + d) * ( 3b/2  b) * (3b/2  b d) and simplifying leads to b = sqrt(2d^2 + sqrt(4d^4 + 16*t^2/3)) simplifying for an equilateral triangle, and letting d = 0, b = 2*t^(1/2)/(3^(1/4)) Thanks for the good answers, all 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Triangle puzzle  MattcAnderson  Homework Help  12  20161108 12:10 
Triangle  davieddy  Puzzles  2  20100629 13:18 
Triangle of Primes  davar55  Puzzles  4  20090702 20:16 
Triangle puzzle  ZetaFlux  Puzzles  12  20070316 19:05 
Triangle puzzle  dsouza123  Puzzles  16  20060526 01:58 