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Given a point outside a circle, two lines can be drawn through that point that
are tangent to the circle. The tangent
segments whose endpoints are the points of tangency
and the fixed point outside the circle are equal. In other words, tangent
segments drawn to the same circle from the same point (there are two for every
circle) are equal.
Figure %: Tangent segments that share an endpoint not on the circle are equal
Chords
Chords within a circle can be related many ways.
Parallel chords in the same circle always cut congruent arcs. That is, the arcs
whose endpoints include one endpoint from each chord have equal measures.
Figure %: Arcs AC and BD have equal measures
When congruent chords are in the same circle, they are equidistant from the
center.
Figure %: Congruent chords in the same circle are equidistant from the center
In the figure above, chords WX and YZ are congruent. Therefore, their distances
from the center, the lengths of segments LC and MC, are equal.
A final word on chords: Chords of the same length in the same circle cut
congruent arcs. That is, if the endpoints of one chord are the endpoints of one
arc, then the two arcs defined by the two congruent chords in the same circle
are congruent.
Intersecting Chords, Tangents, and Secants
A number of interesting theorems arise from the relationships between
chords, secant segments, and tangent segments that intersect. First of all,
we must define a secant segment. A secant segment is a segment with one
endpoint on a circle, one endpoint outside the circle, and one point between
these points that intersects the circle. Three theorems exist concerning the
above segments.
Theorem 1
PARGRAPH
When two chords of the same circle intersect, each chord is divided into two
segments by the other chord. The product of the segments of one chord is equal
to the product of the segments of the other chord.
