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Triangle to Triangle (Posted on 2015-03-07) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Comment 1 of 1
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 
t > (2x^2 - 4xe + e^2) / (3x^2 - 2xe)
The limit of this as e0 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
t < (2x^2-e^2)/(x^2-2xe)
The limit of this as e0 is t<2.

Assuming I didn't miss any cases and we have
but since e cannot actually equal zero we can include the endpoints
2/3 ≤ t 2

  Posted by Jer on 2015-03-09 14:23:21
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