Golden Ratio and Midpoints
In a 2011 article, Jo Niemeyer, offered an elegant way of constructing the Golden Ratio with three equal segments, their midpoints and a pair of perpendicular lines.
Three equal segments A1B1, A2B2, A3B3 are positioned in such a way that the endpoints B2, B3 are the midpoints of A1B1, A2B2 respectively, while the endpoints A1, A2, A3 are on a line perpendicular to A1B1.
In this arrangement, A2 divides A1A3 in the golden ratio, namely,
For a proof, drop a perpendicular B3H from B3 to A1A2.
Assume for convenience that all three line segments are of length 2. Then in right triangle A3B3H,
By the Pythagorean theorem,
(A3H)² = (A3B3)² - (B3H)² = 4 - 1/4 = 15/4,
so that A3H = √15/2.
On the other hand, in right triangle A1A2B2, A2B2 = 2, and
It follows that A1A3 = A1H + A3H = √3(√5 + 1)/2, and, finally,