Show that there cannot be a triangle in which the 3 sides and the area are all primes.
All primes are either odd or 2.
If all three sides are odd, the semiperimeter will end in .5
Using Heron's formula, the product s(s-a)(s-b)(s-c) will end in .0625 and the area will not be a whole number.
If two sides are odd, the third side will be 2. By the triangle inequality, the two odd sides must differ by less than 2 so they are equal. Call this p. The triangle is now isosceles with base 2 so the area = the height. The height is sqrt(p^2-1) which is never a whole number.
If one side is odd, the others are both 2. The odd side must be 3. This triangle has area 3sqrt(7)/4.
If no sides are odd, the triangle is equilateral with sides of 2 and area sqrt(3).
So if all three sides are prime, the area cannot even be a whole number.
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Posted by Jer
on 2024-06-03 11:37:20 |
Solution
