Proof: By Euclid
(related to Proposition: 3.31: Relative Sizes of Angles in Segments)
- Let $AE$ have been joined, and let $BA$ have been drawn through to $F$.
- And since $BE$ is equal to $EA$, angle $ABE$ is also equal to $BAE$ [Prop. 1.5].
- Again, since $CE$ is equal to $EA$, $ACE$ is also equal to $CAE$ [Prop. 1.5].
- Thus, the whole (angle) $BAC$ is equal to the two (angles) $ABC$ and $ACB$.
- And $FAC$, (which is) external to triangle $ABC$, is also equal to the two angles $ABC$ and $ACB$ [Prop. 1.32].
- Thus, angle $BAC$ (is) also equal to $FAC$.
- Thus, (they are) each right angles [Def. 1.10] .
- Thus, the angle $BAC$ in the semicircle $BAC$ is a right angle.
- And since the two angles $ABC$ and $BAC$ of triangle $ABC$ are less than two right angles [Prop. 1.17], and $BAC$ is a right angle, angle $ABC$ is thus less than a right angle.
- And it is in segment $ABC$, (which is) greater than a semicircle.
- And since $ABCD$ is a quadrilateral within a circle, and for quadrilaterals within circles the (sum of the) opposite angles is equal to two right angles [Prop. 3.22] [angles $ABC$ and $ADC$ are thus equal to two right angles], and (angle) $ABC$ is less than a right angle.
- The remaining angle $ADC$ is thus greater than a right angle.
- And it is in segment $ADC$, (which is) less than a semicircle.
- I also say that the angle of the greater segment, (namely) that contained by the circumference $ABC$ and the straight line $AC$, is greater than a right angle.
- And the angle of the lesser segment, (namely) that contained by the circumference $AD[C]$ and the straight line $AC$, is less than a right angle.
- And this is immediately apparent.
- For since the (angle contained by) the two straight lines $BA$ and $AC$ is a right angle, the (angle) contained by the circumference $ABC$ and the straight line $AC$ is thus greater than a right angle.
- Again, since the (angle contained by) the straight lines $AC$ and $AF$ is a right angle, the (angle) contained by the circumference $AD[C]$ and the straight line $CA$ is thus less than a right angle.
- Thus, in a circle, the angle in a semicircle is a right angle, and that in a greater segment (is) less than a right angle, and that in a lesser [segment] (is) greater than a right angle.
- And, further, the [angle] of a segment greater (than a semicircle) [is] greater than a right angle, and the [angle] of a segment less (than a semicircle) is less than a right angle.
- (Which is) the very thing it was required to show.
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Footnotes
