I was thinking about right triangles the other day and wondered what constraints on a triangle's geometry preserve the right angle under rotation. In more visual terms, given the diagram below, what are the relationships between a, b, c, and d that make the angle θ a right angle?
The legs adjacent to θ have the lengths
and the hypotenuse opposite θ has the length
To find the constraints on a, b, c, and d that make the inner triangle a right triangle, we can put these lengths into the Pythagorean theorem:
That is, a is to c as d is to a + b.
If that relationship is a bit abstract, look back at the diagram and note that a and c form the legs of an outer right triange, as do d and a + b. The ratio we found tells us that to make the inner triangle a right triangle, those outer right triangles have to be similar to each other.
That seemed too simple until I remembered the middle-school algebra fact that, given a line, any line perpendicular to it will have a slope that's the negative reciprocal of the slope of the original line. We've lost the negative sign (since directions are implicit in the diagram), but we still have the reciprocal.