Golden section triangle
All triangles that appear after the diagonal of a regular pentagon is full are golden section triangles. ? The golden section triangle has a particularity. All triangles can generate triangles similar to themselves with four congruent triangles, but the golden section triangle is the only triangle that can generate triangles similar to itself with five congruent triangles instead of four congruent triangles. Because the vertex angle of the five-pointed star is 36 degrees, it can also be concluded that the golden section value is 2Sin 18.
As shown in the figure:
Prove: ∫BD= 1/2ab, ∴AB=2BD, so that BD = 1,?
Then AB=2, from Pythagorean theorem:
AD^2=AB^2+BD^2
=2^2+ 1^2=5,∴ad=√5,∵bd=de= 1,ac=ae,
∴ac=ad-de=√5- 1,∵ac=√5- 1,ab=2
∴AC:AB=(√5- 1)/2。