A triangle has area Delta ... ?

The scalene triangle of area Delta has lengths a , b and c such that a in QQ and b^n , c^n notin QQ , AA n in ZZ Does there exist values of a , b and c such that Delta^(2m+1) in QQ for some m in ZZ^+
If not, give a proof

1 Answer
Dean R.
Jul 10, 2018

Here's a right triangle that does the trick:

a=4

b=\sqrt{8-\sqrt{60}}

c=\sqrt{8+\sqrt{60}}

m=0

\Delta=1

Explanation:

I'll assume we get to exclude n=0 as an exponent.

Let's see if we can do it with a right triangle a^2=b^2+c^2 with irrational legs. I think we can stick with m=0. If it's true for m=0 it's true for all m>0 as well.

Delta = 1/2 b c

Let's pick \Delta=1, a=4, abbreviate A=a^2, B=b^2, C=c^2 and solve:

BC=4

B+C=16=A

x^2 - 16 x + 4 = 0

B,C= 8 \pm sqrt{60}

That's a solution. We have

a=4

b=\sqrt{8-\sqrt{60}}

c=\sqrt{8+\sqrt{60}}

\Delta=1

Related questions
Impact of this question
1595 views around the world
You can reuse this answer
Creative Commons License