I enjoy applying vector analysis and complex numbers to geometry problems where possible. I think the following combines the chord, secant, and tangent theorems nicely:

Let XY denote the vector from point X to point Y and
PQ.RS the dot product of vectors PQ and RS.

Theorem:
If points A, B, and E are collinear and points C, D, and E are collinear,
then points A, B, C, and D are concyclic if and only if EA.EB = EC.ED.

Can anybody see a configuration of the points A, B, C, D, and E where the theorem does not hold?

Any help will be greatly appreciated.

Bractals

See

https://www.cut-te-knot.org/pythagoras/PPower.shtml

or search the web for Power of a Point Theorem. If the points are not collinear and EA.EB = EC.ED, then the points are concyclic. If they are collinear, they belong to a circle of infinite radius even without EA.EB = EC.ED. However, (on a line, with P at the origin) xy = zw for great many points.