Butterfly Theorem proof using projective geometry, I can't figure how does the cross ratio be equal in this case:

Question

I am recently self-learning about projective geometry and know very few concepts related to it such as vanishing points and cross ratios

since there aren't much resources about it on YouTube, I cannot fully understand it.

Can anyone explain how does the first line of this proof (the cross ratio line) be true? thank you

My knowledge is rather newbie in projective geometry so it would be much appreciated if anyone has any suggestions on what to refer to, thank you very much.

Answer 1

In the proof, $(AP,AB;AD,AQ)$ refers to the cross ratio of lines $AP,AB,AD,AQ$. This notation is a bit unconventional; usually it would be written as $A(P,B;D,Q)$.

A good reference is Chasles Theorem on Cut-the-knot. It shows how the pencil of lines $AP,AB,AD,AQ$ is congruent to the pencil $CP,CB,CD,CQ$ due to the inscribed angle theorem, leading to equal cross ratios.

If you are unfamiliar with the cross ratio of a pencil of lines, look here.

This equality of cross ratios can be used to characterize all conics, not just a circle. One of the goodies that awaits you if you study projective geometry.