Book III, Proposition 21

In a circle the angles in the same segment equal one another.
# Statement Reason
1. Circle exists. given
2. Angle exists within segment . given
3. Angle exists within segment . given
4.
5.
6.
7.
8.
9.
10.
QED. Therefore in a circle the angles in the same segment equal one another.

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Add to proof Summary
Definition of a point.
Definition of a straight line => A line exists
Definition of a straight line => Choose an arbitrary point
Definition of a circle
Definition of an equilateral triangle.
Draw a straight line from any point to any point.
Extend a finite straight line continuously in a straight line.
Draw a circle with center and radius.
All right angles equal one another.
The Parallel Postulate
Things equal to the same thing are equal to each other (lines).
Things equal to the same thing are equal to each other (parallelogram and triangles).
Things equal to the same thing are equal to each other (double angles).
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things which coincide with one another equal one another.
The whole is greater than the part.
Construct an equilateral triangle on a given finite straight line.
Place a straight line equal to a given straight line with one end at a given point.
Cut off from the greater of two given unequal straight lines a straight line equal to the less.
SAS - If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end.
If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.
In a parallelogram, the diameter bisects the areas.
Triangles on the same base and in the same parallels equal one another.
Find the center of a circle.
The angle at the center of a circle is double the angle at the circumference when the angles have the same circumference as base.