When two chords intersect inside a circle, there is a relationship between the lengths of their segments.
Intersecting chords theorem:
If two chords $AC$ and $BD$ intersect at point $P$ inside a circle: $$AP \times PC = BP \times PD$$
Example: Two chords intersect at $P$. One chord has segments of length 3 and 8. The other has one segment of length 4. Find the other segment. $$3 \times 8 = 4 \times x \Rightarrow x = 6$$
Intersecting secants (from external point): If two secants are drawn from an external point $P$: $$PA \times PB = PC \times PD$$ where $A$, $B$ are where one secant meets the circle, and $C$, $D$ the other.
Tangent-secant from external point: $$PT^2 = PA \times PB$$ where $PT$ is a tangent and $PA$, $PB$ are the two intersections of a secant.
These results follow from the fact that the triangles formed are similar (equal angles from circle theorems).
Common error: multiplying the wrong pairs — always multiply the two segments of the same chord (or secant), not across chords.
Circle theorems Ratios and similar shapes
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