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#

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