I would like to construct a regular right angled hexagon in a klein model.
I'm having a hard time understanding why this method works, here is what my professor did in class. Any additional comments will be greatly appreciated. Thanks in advance!
First in a circle O, pick any six points, $A,B,C,D,E,F$, connect $AB$, $CD$, and $EF$.
Then, construct line $a, b, c, d, e, f$ where $a,b$ intersects, $c,d$ intersects, and $e,f$ intersects, call them points $AA, BB,$ and $CC$.
Lastly, connect $AA, BB, CC$ to construct lines $\alpha, \beta,$ and $\gamma$, the hexagon inscribed in the
circle hexagon ($\theta1 - \theta6$) all have right angles therefore is a regular hexagon.