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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. |
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Intersecting Chords Rule:
(segment piece)×(segment piece) =
(segment piece)×(segment piece) |
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Theorem Proof:
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Statements |
Reasons |
| 1. |
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1. |
Given |
| 2. |
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2. |
Two points determine only one
line. |
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3. |
If two inscribed angles
intercept the same arc, the angles are congruent. |
| 4. |
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4. |
AA - If two angles of one
triangle are congruent to the corresponding angles of
another triangle, the triangles are similar. |
| 5. |
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5. |
Corresponding sides of similar
triangles are in proportion. |
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6. |
In a proportion, the
product of the means equals the product of the extremes. |
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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. |
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Secant-Secant Rule:
(whole secant)×(external part) =
(whole
secant)×(external part) |
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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. |
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Secant-Tangent Rule:
(whole secant)×(external part) =
(tangent)2 |
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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:
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