Prove that triangle CBM is similar to triangle BAN, providing detailed steps and justifications using geometric theorems.

Here are the proofs for the given statements:

10.2.3 Proof: $\triangle CBM \text{ ||| } \triangle BAN$

Step 1: Identify equal angles using the tangent-chord theorem.

• AD is a tangent to circle M at A, and AB is a chord of circle M. By the tangent-chord theorem, the angle between the tangent and the chord is equal to the angle in the alternate segment. $$\angle DAB = \angle ACB$$
• AC is a tangent to circle N at A, and AB is a chord of circle N. By the tangent-chord theorem, the angle between the tangent and the chord is equal to the angle in the alternate segment. $$$$