geometry - Found a New Golden Ratio Construction with Equilateral Triangle, Square, and Circle. Geometric/Trigonmetric proof?

The below figure discloses a new golden ratio construction with an equilateral triangle, square, and circle. Geogebra gave me the value of the golden number 1.618 for the ratio of the yellow line to the blue line.

Has anyone seen anything like this construction? I haven't yet.

Here is how I constructed it. If you see an easier way, please let me know!

1. Draw an equilateral triangle.


2. Draw a square with its diagonal one side of the equilateral triangle.


3. Draw a circle centered on the tip of the equilateral triangle with its radius equal to the side of the square.


4. A line drawn through the midpoints of the equilateral triangle will be cut at the golden section. Geogebra states that the ratio of the yellow line to blue line is the golden number 1.618:1.

My first question is can anyone see an easier way to construct this without the square, but with another circle or triangle perhaps? As one can see in the bottom figure, the square is not needed for the final golden ratio representation.

Secondly, does anyone see any quick/easy geometric or trigonometric proofs?

And lastly, if you have seen anything similar, I'd love to see it! Haven't seen this one yet. :)

Thanks!

Answer

Here's a short coordinate proof:

Suppose the triangle has side length $2$, one vertex lies at the origin, and the right half of the $x$-axis bisects the triangle. The circle has radius $\sqrt{2}$, so its Cartesian equation is $x^{2} + y^{2} = 2$. The yellow/blue chord lies on $x = \frac{1}{2}\sqrt{3}$, which intersects the circle at $\frac{1}{2}(\sqrt{3}, \pm\sqrt{5})$. The yellow segment has length $1$, the blue segment has length
$$
\tfrac{1}{2}(\sqrt{5} - 1) = \frac{1}{\frac{1}{2}(\sqrt{5} + 1)},
$$
i.e., $\dfrac{\text{yellow}}{\text{blue}} = \frac{1}{2}(\sqrt{5} + 1)$.