I used David E. Joyce's compilation of Euclid's Elements as my primary source.
All figures and manipulatives were made using GeoGebra.
Definitions
Preliminary Definitions (not in the Elements)
- A chord is a line segment whose endpoints fall on a circle.
- A diameter is a chord that passes through the center of the circle.
- An arc is a portion of a circle with two endpoints.
Definition 1
Circles with equal radii are congruent.
Definition 2
A line is tangent to a circle if it intersects the circle at exactly one point.
Definition 3
Circles are tangent to one another if they intersect in exactly one point. Two circles can be externally tangent or internally tangent.
Definition 4
Two or more chords drawn in a circle are equidistant from the center of the circle if the segments drawn from the center, perpendicular to each chord, are congruent.
Definition 5
A chord is at a greater distance from the center of the circle if the segment drawn from the center, perpendicular to that chord, is longer than the same for another chord.
Definition 6
A segment of a circle is the area enclosed between a chord and the circle containing it.
Definition 7
The angle of a segment is the angle formed by the segment’s chord and a line tangent to the circle at the chord’s endpoint.
Definition 8
An angle in a segment is an angle with vertex on the arc of the segment with sides drawn to the endpoints of the segment’s chord.
Definition 9
The angle in a segment stands upon the arc opposite the vertex of the angle with endpoints at the sides of the angle.
Definition 10
A sector of a circle is the area enclosed between two radii of the circle and the arc between them.
Definition 11
Segments of circles are similar if the angles of the segments are congruent, or if the angles in the segments are congruent.
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Propositions
- Click on the heading for each proposition to go to a GeoGebra manipulative or step-by-step how-to guide.
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How to locate the center of a circle.
The perpendicular bisector of any chord in a circle will pass through the center of the circle.
The segment drawn between any two points on a circle will lie entirely within the circle.
A chord passing through the center of a circle (a diameter) will bisect a chord that does not pass through the center if and only if they are perpendicular to one another.
If two chords in the same circle do not pass through its center, then they do not bisect one another.
If two circles intersect one another, then they are not concentric.
If two circles are tangent to one another, then they are not concentric.
Choose a point on a given diameter of a circle, but not the center. Then consider all of the line segments with this endpoint and another endpoint on the circle. Then
- the longest of these line segments is the one that passes through the center of the circle.
- the shortest of these line segments is the portion of the given diameter that does not include the center.
- among all other line segments drawn from the point to the circle, the ones closer to the line segment that passes through the center are longer, and the ones further from the line segment through the center are shorter.
- these line segments drawn from the point but not along the diameter come in congruent pairs, on opposite sides of the diameter.
Choose a point outside of a circle. First, consider all of the chords that could be drawn within the circle on a line through this point. Then
- the chord that passes through the center of the circle is the longest.
- among all of the other chords, the ones closer to the chord that passes through the center are longer than ones that are further from it.
Now consider all of the line segments that could be drawn between the point outside of the circle and a point on the circle without passing through the interior of the circle. Then
- the line segment that falls on a line between the point and the center of the circle is the shortest.
- among all other line segments drawn as described, the ones closer to the shortest line segment are shorter than the ones further from it.
- except for the shortest one, these line segments come in congruent pairs, on opposite sides of the shortest line segment.
If from a point inside a circle three or more congruent line segments can be drawn between the point and the circle at one time, then the point is the center of the circle.
Two circles can intersect one another in at most two points.
If two circles are internally tangent, then the line that passes through both of their centers will also include the point of tangency of the circles.
If two circles are externally tangent, then the line that passes through their centers will also include the point of tangency of the circles.
Tangent circles intersect in exactly one point, whether they are internally or externally tangent.
Two chords in the same circle are congruent if and only if they are the same distance from the center.
Among all of the chords in a given circle, the diameter is the longest. Among all other chords, one that is closer to the center is longer than one that is further from the center.
A line drawn perpendicular to a diameter of a circle at one endpoint will fall entirely outside of the circle. At the same point, it is not possible to construct a second distinct line that does not pass through the circle.
Corollary
A line perpendicular to a diameter of a circle at one endpoint is tangent to the circle.
How to draw a line tangent to a circle from a given point.
A line tangent to a circle is perpendicular to the radius at the point of tangency.
A line drawn perpendicular to a tangent line at the point of tangency will pass through the center of the circle.
The the measure of an arc’s central angle is double that of an angle with its vertex on the portion of the circle opposite the arc.
Any two angles in the same segment (see Definition 8) are congruent.
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
On the same side of one line segment, it is not possible to construct two distinct segments that are similar (see Definition 11) but not congruent.
Similar segments (see Definition 11) of circles constructed on congruent line segments are congruent.
Given a segment of a circle, how to construct the circle that contains it.
In congruent circles, congruent central angles will subtend congruent arcs, and so will congruent inscribed angles.
Proposition 27 (Converse of Proposition 26)
In congruent circles, central angles that subtend congruent arcs are congruent, and so are inscribed angles that subtend congruent arcs.
In congruent circles, congruent chords cut the circle into two pairs of congruent arcs. The greater arcs are congruent to one another and the lesser arcs are congruent to one another.
Proposition 29 (Converse of Proposition 28)
In congruent circles, chords that cut off congruent arcs are congruent.
How to bisect a given arc.
An angle inscribed in (see Definition 8) a semicircle is a right angle.
An angle inscribed in a segment larger than a semicircle is acute.
An angle inscribed in a segment smaller than a semicircle is obtuse.
Furthermore, the angle of (see Definition 7) a segment larger than a semicircle is obtuse, and the angle of a segment smaller than a semicircle is acute.
The angles between a chord and a tangent line at one of its endpoints are equal to the angles in (see Definition 8) the opposite segments divided by the chord.
On a given line segment, how to construct a segment of a circle so that the angle in (see Definition 8) the segment is the same as a given angle.
Given a circle, how to cut off a segment so that the angle in (see Definition 8) the segment is the same as a given angle.
If two chords in the same circle intersect one another, then the rectangle formed using the two smaller pieces of one of those chords as side lengths will have the same area as the rectangle formed from the pieces of the other chord.
Consider a point outside of a circle. Suppose two line segments share this as one endpoint and have other endpoints on the circle so that one of them is tangent to the circle and the other one passes through the circle. Then the area of the square formed using the tangent line segment’s length will be equal to the area of the rectangle formed using as side lengths the whole of the other line segment and the portion of that segment that lies outside of the circle.
Proposition 37 (Converse of Proposition 36)
Consider a point outside of a circle. Suppose two line segments share this endpoint and have their other endpoints on the circle so that one of them passes through the circle and the other does not. If the area of the square formed using the length of the segment that does not pass through the circle is equal to the area of the rectangle formed using as side lengths the whole of the other line segment and the portion of that segment that lies outside of the circle, then the line segment that does not pass through the circle is tangent to it.
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