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Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle

(Proposition 16 from Book 3 of Euclid's “Elements”)

A (straight line) drawn at right angles to the diameter of a circle, from its end, will fall outside the circle. And another straight line cannot be inserted into the space between the (aforementioned) straight line and the circumference. And the angle of the semicircle is greater than any acute rectilinear angle whatsoever, and the remaining (angle is) less (than any acute rectilinear angle).

• Let $ABC$ be a circle around the center $D$ and the diameter $AB$.
• I say that the (straight line) drawn from $A$, at right angles to $AB$ [Prop. 1.11], from its end, will fall outside the circle.

Modern Formulation

A straight line ($AE$) going through the endpoint ($A$) of a diameter ($\overline{AB}$) of a given circle is perpendicular to this diameter and lies outside the circle.

Moreover:

• if $\alpha$ is the angle¹ between $AE$ and the line of the circumference of the circle at the point $A,$
• and if $\beta$ is the angle¹ between the diameter $\overline{AB}$ and the line of the circumference of the circle at the point $A$.

Then for any acute rectilinear angle $\gamma$ the following holds: $\alpha < \gamma < \beta.$

Table of Contents

Proofs: 1 Corollaries: 1

Mentioned in:

Proofs: 1 2 3 4

Thank you to the contributors under CC BY-SA 4.0!

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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

Footnotes