Proof that the Golden Ratio is in the 3-4-5 Right Triangle

In my previous post Golden Rectangles in a Right Triangle I inscribed circles in a right-angled triangle with sides of length 3, 4, and 5. I measured that the radius of one inscribed circle was 1/φ². This was an empirical observation of limited accuracy, not a mathematical proof. Phi or φ, equal to (1+√5)/2, is a transcendental number. Note Paul’s correction in the comments : phi is not transcendental. Fortunately, my colleague Ed Staples was able to prove the length of the radius using trigonometry. His work is shown in the figure. The length r to be calculated is the radius of the small circle inscribed in the corner between the sides of length 3 and length 5. He proves that r = 2 – φ. And, 2 – φ is equal to 1/φ², which is the value that was determined approximately from the construction. I used the Geogebra software to make this construction. The software conveniently measures lengths and angles, which was particularly useful in this case. You can find Ed Staples’ blog, with a variety of mathematical puzzles and tutorials, at https://www.mathematicalwhetstones.com/blog