If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will be on the straight line so drawn.
Proof
By III.18, the line from the centre to the contact point is
perpendicular to the tangent. Conversely, the perpendicular at the
contact point in the plane is unique (I.11), so the centre must lie
on it. (If the centre were off this perpendicular, the line from
centre to contact would not be perpendicular to the tangent,
contradicting III.18.)