Thunderbolt: This post is about that thing the geometry pros call a "harmonic quadrilateral":

are cyclic and

If you connect each vertex to any point on the circle (draw a tangent if the point coincides with a vertex) you get a harmonic pencil, and vice-versa. (Proof: trig form of harmonic pencils, or invert)

Also the second half of this post involves a symmedian whose intersection with the circle would also form a harmonic quadrilateral. (Proof: ratio chase)

Projective geometry notes:

collinearity, cross-ratio are preserved, therefore so are harmonic divisions/pencils/quadrilaterals
projective mappings can send any four noncollinear points to any other four noncollinear points (including points at infinity! yay!)
theorems: Pascal (on conics (+ degenerate cases, esp. Pappus)), Brianchon, Desargues
perfectly bijective pole/polar correspondence
points correspond to lines through a point in space, just pick a plane and take intersections (and all of a sudden, we have a reason to think of conic sections as conic sections and not as random loci or quadratic equations. They map to each other. Move the plane around. Wow.)

This stuff always seems overpowered until you remember that problems like the one where you saw this used are about 1% of geometry problems. Oh well.

Hey wait I still haven't posted any algebra since the renovation...