Rules for Dealing with Chords, Secants, Tangents in Circles
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Theorem 1:

If two chords intersect in a circle, the product of the lengths of the segments of one chord equal the product of the segments of the other.

 
Intersecting Chords Rule:
(segment piece)×(segment piece) =
              (segment piece)×(segment piece)

 

Theorem Proof:

Statements Reasons
1. 1. Given
2.   2. Two points determine only one line.
3.   3. If two inscribed angles intercept the same arc, the angles are congruent.
4.   4.  AA - If two angles of one triangle are congruent to the corresponding angles of another triangle, the triangles are similar.
5. 5. Corresponding sides of similar triangles are in proportion.
6.   6.  In a proportion, the product of the means equals the product of the extremes.

 

Theorem 2:

If two secant segments are drawn to a circle from the same external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part.

 
Secant-Secant Rule:
(whole secant)×(external part) =
            (whole secant)×(external part)

 

 

Theorem 3:

If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.

 
Secant-Tangent Rule:
(whole secant)×(external part) =
                                                 (tangent)2

 

This theorem can also be stated as "the tangent being the mean proportional between the whole secant and its external part" (which yields the same final rule: