## Special isosceles triangles

### Problem 138

Published on Saturday, 20th January 2007, 11:00 am; Solved by 3512; Difficulty rating: 45%Consider the isosceles triangle with base length, *b* = 16, and legs, L = 17.

By using the Pythagorean theorem it can be seen that the height of the triangle, *h* = √(17^{2} − 8^{2}) = 15, which is one less than the base length.

With *b* = 272 and L = 305, we get *h* = 273, which is one more than the base length, and this is the second smallest isosceles triangle with the property that *h* = *b* ± 1.

Find ∑ L for the twelve smallest isosceles triangles for which *h* = *b* ± 1 and *b*, L are positive integers.