Proposition III.19 — If a straight line touch a circle, and from the point of con…

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.)

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Depends on (3)

III.18Proposition III.18If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight…
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
III.2Definition III.2A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

Required by (dependents) (1)

III.37Proposition III.37If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut…

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