The Secant Theorem
- Author:
- Ryan Hirst
- Topic:
- Circle, Geometry, Tangent Line or Tangent
The Secant theorem relates two secants of a circle by the the point of intersection of the chords. It is used to define
the power of a circle at a point.
PA * PB = PC * PD is true for any location of P.
However,
If P is outside,
|PB - PA| = AB (secant 1)
|PD - PC| = CD (secant 2)
P inside:
PB + PA = AB
PD + PC = CD
P on the circle:
PA*PD = PC*PD = 0
(One segment in each multiplication will always be zero.)
If I want to assign meaning to this relationship, I should take the difference of sign into account.
For example, the power of a point is negative if P is inside the circle.
Onward.