TITLE: How do I prove that $\left( 1-\frac{a}{b} \right)\left( 1+\frac{c}{d} \right)=4$?
QUESTION [20 upvotes]: The challenge is to prove $$\left(1-\dfrac{a}{b}\right)\left(1+\dfrac{c}{d}\right)=4.$$

Apart from the Pythagorean theorem, the tangent and secant theorem, and the cosine theorem, I could not invent anything. The calculations here turn out to be very cumbersome and I'm not sure that this is how it should be initially solved.
I had an idea to start from the similarity in the figure, there is a lemma about a right angle on a chord, there is a property of a tangent and a secant...But all this again will lead to cumbersome calculations.

REPLY [5 votes]: So what I want to say is that I found what could be said to be the "best" way to solve this problem. First, consider the following drawing:

All colored $\triangle$ are pairwise similar to each other, $\left(r_{a} r=a b\right)$.
Yellow $\sim$ blue $\Rightarrow \dfrac{b-a}{2 r}=\dfrac{B_{2} K_{2}}{c+d}$.
Blue $\sim$ red $\Rightarrow \dfrac{B_{1} K_{1}}{d}=\dfrac{2 r}{a}$.
Blue $\sim$ green $\Rightarrow \dfrac{B_{2} K_{2}}{B_{1} K_{1}}=\dfrac{r_{a}}{r}$.
We get:
$$ (b-a)(c+d)=2 B_{2} K_{2} \cdot r=2 B_{1} K_{1} \cdot r_{a}=4 r_{a} r \frac{d}{a}=4 b d $$