Determine all real constants t such that whenever a, b and c are the lengths of sides of a triangle, then so are a^2+bct, b^2+cat, c^2+abt.
The two most extreme cases are
nearly co-linear equally spaced points
a=b=2c-e (where e is a tiny quantity)
and a long narrow isosceles triangle
a=b, c=e (where e is a tiny quantity)
case 1.
a=x, b=x, c=2x-e
a^2+bct = x^2 + (2x^2 + xe)t
b^2+cat = x^2 + (2x^2 + xe)t
c^2+abt = (4x^2 - 4xe + e^2) + (x^2)t
By the triangle inequality
x^2 + (2x^2 + xe)t + x^2 + (2x^2 + xe)t > (4x^2 - 4xe + e^2) + (x^2)t
or
t > (2x^2 - 4xe + e^2) / (3x^2 - 2xe)
The limit of this as e→0 is t>2/3
case 2.
a=b=x, c=e
a^2+bct = x^2 + (xe)t
b^2+cat = x^2 + (xe)t
c^2+abt = e^2 + (x^2)t
By the triangle inequality
2x^2 + (2xe)t > e^2 + (x^2)t
or
t < (2x^2-e^2)/(x^2-2xe)
The limit of this as e→0 is t<2.
Assuming I didn't miss any cases and we have
2/3<t<2
but since e cannot actually equal zero we can include the endpoints
2/3 ≤ t ≤2
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Posted by Jer
on 2015-03-09 14:23:21 |
Solution