New Golden Ratio Conjecture with Triangle and Square: It is very close, but is it really the golden ratio?

Question

Geogebra gives me 1.616 for the ratio of the blue segment p to the red segment q instead of the golden ratio 1.618 for the construction shown below, so it could be close to PHI, but no cigar.

This construct is created by drawing a square and then drawing an equilateral triangle with one vertex touching the side of the square as drawn.

A diagonal (segment n) is then drawn from D to F, cutting the square at H.

Segment q extends from C to H and segment extends from C to G.

Is the ratio of the blue segment p to the red segment q the golden ratio? I have my doubts, but it is so very close.

Thanks and best regards! :)

Answer 1

From your diagram

• C seems to be at $(4,4)$
• G seems to be at $(4+2\sqrt3,4)$
• H seems to be at $(4,-12+8\sqrt3 )$

Which makes $p=2\sqrt3$ and $q=16-8\sqrt3$ and $\dfrac{p}{q}=\dfrac34+\dfrac{\sqrt3}2 \approx 1.616025$ which is slightly less than the golden ratio $\dfrac12+\dfrac{\sqrt5}2 \approx 1.618034$