# Pythagorean Theorem Converse

One of the joys of getting old is that you forget things.   So one of the things I recall is that the converse of the Pythagorean Theorem is true, that is, if a triangle with short sides a and b and long side c is such that

a2 + b2 = c2,

then the triangle must be a right triangle with the angle between sides a and b being 90°.  But I didn’t recall how to prove it.  So I thought I would see if I could do it without looking up any sources.

See the Pythagorean Theorem Converse

# Magic Pythagorean Circle

This statement showed up recently at Futility Closet and I found it to be another one of those magical results that seemed so surprising. I don’t recall ever seeing this before.

“The radius of a circle inscribed in a 3-4-5 triangle is 1.
(In fact, the inradius of any Pythagorean triangle is an integer.)”

(A Pythagorean triangle is a right triangle whose sides form a Pythagorean triple.) Futility Closet left these remarkable statements unproven, so naturally I felt I had to provide a proof.

# Right Triangle with Roots

This is an interesting problem from the United Kingdom Mathematics Trust (UKMT) Senior Math Challenge of 2008.

“The length of the hypotenuse of a particular right-angled triangle is given by √(1 + 3 + 5 + … + 23 + 25). The lengths of the other two sides are given by √(1 + 3 + 5 + … + (x – 2) + x) and √ (1 + 3 + 5 + … + (y – 2) + y) where x and y are positive integers. What is the value of x + y?”