Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular the horizontal line. One end of the green line is attached to the red line's midpoint, while the other end of the green line touches the horizontal line. One end of orange line is attached to the midpoint of the green line, while the other end of the orange line touches the horizontal line. This arrangement splits the horizontal line in a manner corresponding to the golden ratio, with the red line being located at the golden cut of the horizontal line.
Via geogebra I was able to determine that the ratio of the yellow line to the blue line is the golden section. It gave me a number of 1.619... which is close enough I believe to the golden ratio ~1.618...
Then, I thought I saw a circle implied by the arrangement. Curiously enough, the radius of the circle in the lower figure seems to be exactly 3/4 of the length of one of the lines. Is there some way to prove this in geogebra?
I really do think that geogebra allowed me to determine that the horizontal line is cut into the golden ratio of yellow and blue lengths by the red line. And it looks like the entire golden ratio construct can be naturally inscribed in a circle with a radius of 3/4 of one of the lines. Are there any other cool relationships this implies?
Thanks again for all the help, and thanks for the geogebra tip from Blue. I look forward to working with it!