21Bcircletheorems

= Circle Geometry Theorems =

Intersecting Chords, Secants and Tangents

 * Theorem 6 : Intersecting Chords **
 * If two chords ** AB ** and ** CD ** intersect inside a circle at a a point ** P **, then
 * ** AP × PB = CP × PD **
 * If we label the chords as
 * AB = a + b
 * CD = c + d
 * then
 * ** a × b = c × d **
 * In the example shown
 * 4 × 3 = 6 × 2


 * Theorem 7 : Intersecting Secants **
 * If two secants ** AP ** and ** CP ** intersect outside the circle at a point ** P **, then
 * ** AP × BP = CP × DP **
 * If we label the secants as
 * AP = a, .. BP = b
 * CP = c, .. DP = d
 * then
 * ** a × b = c × d **
 * In the example shown
 * Total length of purple secant is 7 + 1 = 8
 * Total length of green secant is 2 + 2 = 4
 * 8 × 1 = 4 × 2


 * Theorem 8: Intersecting Tangents **
 * If two tangents AP and BP intersect at a point P, then
 * ** AP = BP **
 * or
 * ** a = b **


 * Theorem 9: Radius bisecting Chord **
 * If a radius OX and a chord AB intersect at right angles at a poiint P, then
 * the radius ** bisects **the chord (__**bisect**__ means cuts it exactly in half)
 * ** AP = PB **
 * or
 * ** a = b **
 * The reverse is also true:
 * If a radius __**bisects**__ a chord, then they intersect at right angles.
 * ** OP **** ^ **** AB ** .. ** Û ** .. ** AP = PB **




 * Theorem 10: Chords of equal length **
 * If two chords AB and CD are equal in length, then
 * they are both the same distance from the centre.
 * ** OX = OY ** .. ** Û ** .. ** AB = CD **
 * The reverse is also true
 * If two chords are the same distance from the centre, then they are equal in length.


 * Note: **
 * There is nothing special about the numbers assigned to these theorems
 * The numbers are simply the order we are listing the theorems.

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