TITLE: Is there a Pythagorean triple whose angles are 90, 45, and 45 degrees?
QUESTION [11 upvotes]: Is there a Pythagorean triple (a.k.a. an integer triangle) whose angles are 90, 45 and 45 degrees? I am trying to connect LEGO roads at angles other than the standard 90 degrees.

REPLY [14 votes]: In your context you might be interested in isosceles triangles that are almost right.  As others have said, a right isosceles triangle has sides that are $a,a,a\sqrt 2$ and as $\sqrt 2$ is not rational we cannot have an integer sided one.  However, if we find a rational number that is close to $\sqrt 2$ we can find isosceles triangles that are close to right.  We have $\sqrt 2 \approx 1.414213$, while $\frac 75 = 1.4$ is not so far away, so a $5,5,7$ triangle is close to right.  In fact the angle is $\arccos \left(\frac 1{50}\right)\approx 88.85^\circ$.  You might have enough give to tolerate that.  If not, given one triangle in the list is $a,a,b$ the next is $a+b,a+b,2a+b$, so the next is $12,12,17$, then $29,29,41$, and so on.  The get closer and closer to right as you progress.  If you are interested in where this comes from, you could look up Pell's equation.