314 The golden section
10.4 The golden right triangle
The golden right triangle is the one whose hypotenuse and one leg are in
the golden ratio.
B
C
A 1
φ
Characterizations of the golden triangle
1.If the sides of a right triangle are in geometric progression, then it
is the golden triangle.
2.If the altitude of a right triangle divides the hypotenuse in the golden
ratio, then the triangle is golden.
3.If one side of a right triangle is the geometric mean of the other two
sides, then this is a golden right triangle.
4.If the sides of a right triangle are respectively the harmonic, geo-
metric, and arithmetic means of two numbers, then the triangle is a
golden right triangle.