Suppose line m intersects ⊙X at point Z and m is perpendicular to XZ. What is the maximum number of other points on ⊙X that m can intersect?

The Perpendicular Tangent Theorem tells us that in the situation described above, line m must be tangent to ⊙X at Z. By definition of tangent line, line m must intersect ⊙X in exactly one point. That means that the maximum number of points other than P on ⊙X that m can intersect is zero.