138.rb

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Special isosceles triangles
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 = √(172 − 82) = 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.



h = sqrt(l**2 - (b/2)**2) = (b-1)

Let k = b/2


sqrt(l**2 - k**2) = 2k + 1
 


Math.sqrt((l+k)(l-k)) - 2k - 1 = 0



sqrt(l+k) * sqrt(l-k) = 2k + 1



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