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Special isosceles triangles

Published on Saturday, 20th January 2007, 11:00 am; Solved by 5047;
Difficulty rating: 45%

Problem 138

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, $h = \sqrt{17^2 - 8^2} = 15$, 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 $h = b \pm 1$.

Find $\sum L$ for the twelve smallest isosceles triangles for which $h = b \pm 1$ and $b$, $L$ are positive integers.