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Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center

(Proposition 19 from Book 3 of Euclid's “Elements”)

If some straight line touches a circle, and a straight line is drawn from the point of contact, at right-[angles to the tangent, then the center (of the circle) will be on the (straight line) so drawn.

• For let some straight line $DE$ touch the circle $ABC$ at point $C$.
• And let $CA$ have been drawn from $C$, at right angles to $DE$ [Prop. 1.11].
• I say that the center of the circle is on $AC$.

Modern Formulation

If a straight line ($AC$) is perpendicular to a tangent ($DE$) and goes through the point ($C$) at which the tangent touches the circle, then the straight line also goes through the center of the circle.

Table of Contents

Proofs: 1

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Proofs: 1

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016