000RM.dvi

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.