A circle centered on side AC of triangle ABC passes through vertex C

A circle centered on side AC of triangle ABC passes through vertex C and touches line AB at point B. Find the diameter of the circle if AB = 9, AC = 12.

Let the circle intersect the side AC at point K, then KC is the diameter.
According to the theorem on the tangent and secant: if from one point (A) the tangent (AB) and secant (AC) are drawn to the circle, then the product of the entire secant (AC) by its outer part (AK) is equal to the square of the tangent (AB). Then:
AB ^ 2 = AK * AC;
12AK = 9 ^ 2;
12AK = 81;
AK = 81/12;
AK = 6.75 conventional units.
Point K divides the AC into two segments:
AC = AK + KC;
12 = 6.75 + KC;
KC = 12 – 6.75;
KC = 5.25 conventional units.
Answer: KC = 5.25 conventional units.