— GEOMETOR • elements

G.61 diameter of a circle

A diameter of a circle is any segment drawn through the center and terminated in both directions by the arc of the circle; it bisects the circle.

G.96 semicircle

A semicircle is the figure contained by a diameter and the arc cut off by it.

G.99 chord of a circle lies within the circle

The segment joining any two points on the arc of a circle falls within the circle.

G.106 distances to a circle from an exterior point

From a point outside a circle, of segments falling on the concave arc the greatest passes through the center and nearer ones are greater, while on the convex arc the least lies between the point and the diameter and nearer ones are less, with at most two equal segments.

G.117 angle of a circle segment

An:term:**angle of a segment** <angle-of-segment> is that contained by a:term:segment <G.4> and a arc of a:term:circle <G.10>.

G.118 perpendicular at extremity of a diameter is tangent

The line drawn at right angles to the diameter of a circle from its extremity falls outside the circle, no other line can be interposed between it and the arc, and the angle of the semicircle is greater than any acute rectilineal angle.

G.122 sector of a circle

A:term:**sector of a circle** <sector-of-circle> is the:term:figure <G.9> which, when an:term:angle <G.22> is constructed at the:term:center <center> of the:term:circle <G.10>, is contained by the:term:segments <G.4> containing the angle and the arc cut off by them.

G.123 inscribed angle theorem

In a circle, the angle at the center is double the angle at the circumference when the angles have the same arc as base.

G.126 segment of a circle

A:term:**segment of a circle** <segment-of-circle> is the:term:figure <G.9> contained by a:term:segment <G.4> and a arc of a:term:circle <G.10>.

G.132 angle standing on an arc

And, when the:term:segments <G.4> containing the:term:angle <G.22> cut off a arc, the angle is said to:term:**stand upon** <stand-upon-arc> that arc.

G.133 equal angles stand on equal arcs

In equal circles, equal angles stand on equal arcs, whether they stand at the centers or at the circumferences.

G.134 angles standing on equal arcs are equal

In equal circles, angles standing on equal arcs are equal to one another, whether they stand at the centers or at the circumferences.

G.135 equal chords cut off equal arcs

In equal circles, equal chords cut off equal arcs, the greater equal to the greater and the less to the less.

G.136 equal arcs are subtended by equal chords

In equal circles, equal arcs are subtended by equal chords.

G.137 bisect an arc

Bisect a given arc of a circle.

G.138 angle in a circle segment

An:term:**angle in a segment** <angle-in-segment> is the:term:angle <G.22> which, when a:term:point <G.1> is taken on the arc of the segment and:term:segments <G.4> are joined from it to the:term:extremities <G.6> of the segment which is the:term:**base of the segment** <base-of-segment>, is contained by the segments so joined.

Posts tagged 'arc'