In this visual proof for the law of cosines, why are the products of subsegments of two intersecting chords equal?

Question

The first line of the visual proof below states that

$$(2a\cos\theta-b)b=(a-c)(c+a)$$

I understand the line segments represented by each part of the equation, but what makes the equation true?

In other words, what makes the product of line segments $2 a\cos\theta – b$ and $b$, equivalent to the product of line segments $a – c$ and $c+a$?

Answer 1

It's called the "intersecting chords theorem" (not surprisingly) and is written out here, with proof:

https://en.wikipedia.org/wiki/Intersecting_chords_theorem