Bidirectionally of the "Tangent Criterion"

Question

I've recently been reviewing some basic geometry concepts when I saw this one in Evan Chen's fantastic "Euclidean Geometry in Mathematical Olympiads" (EGMO).

Proving $(i)\Rightarrow (iii)$ is quite simple.

Hint: Move point $C$ in the circumcircle so that $\angle BAC=90°$

Nevertheless, I've had some issues trying to prove that this proposition is biconditional, i.e. $(i)\iff (iii)$.

Since one of the directions is already proven, I only need to show $(iii)\Rightarrow (i)$

What would you suggest?

Answer 1

Unsurprisingly, the solution is

Move point $C$ in the circumcircle so that $\angle ABC=90°$.