Proof
Let be a diameter of the circle with centre , and any
point on the circle other than , . Join , , .
Since (radii), is isoceles, so by I.5,
. Similarly is isoceles
with . By I.32, the angles of
sum to two right angles:
\[
\angle OAC + \angle OBC + \angle ACB \;=\; \text{two right angles.}
\]
But
by the isoceles equalities. Substituting,
two right angles, so is right.
For inscribed angles in segments greater than a semicircle, the
arc is less than a semicircle, so by III.20 the inscribed angle is
half a central angle less than two right angles — hence less than
a right angle. The lesser-segment case is symmetric.
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Full neighborhood
Depends on (6)
- I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal straight lines be produced…
- I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
- III.20Proposition III.20In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same…
- 2Common notion 2If equals be added to equals, the wholes are equal.
- 3Common notion 3If equals be subtracted from equals, the remainders are equal.
- I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…
Required by (dependents) (4)
- III.16Proposition III.16The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle,…
- III.32Proposition III.32If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line…
- IV.6Proposition IV.6In a given circle to inscribe a square.
- VI.13Proposition VI.13To two given straight lines to find a mean proportional.
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