This document discusses the product of intercepts theorem as it relates to chords and secants of a circle. It states that the product of the intercepts formed by two intersecting chords or secants on the circle's circumference is equal to the product of the intercepts formed at the other two points. This relationship is illustrated with diagrams and the theorem is expressed algebraically as AX × BX = CX × DX. It also notes that for secants, the intersection occurs outside the circle and introduces the related theorem that the square of a tangent equals the product of its intercepts.

1. Products of Intercepts

2. Products of Intercepts
D
B
X
A
C

3. Products of Intercepts
D
B
AX  BX  CX  DX
X
A
C

4. Products of Intercepts
D
B
AX  BX  CX  DX
X product of intercepts of intersecting chords
A
C

5. Products of Intercepts
D
B
AX  BX  CX  DX
X product of intercepts of intersecting chords
A
C
“endpoint of the chord to the point of intersection (AX) times endpoint
of the chord to the point of intersection (BX) equals endpoint of the
chord to the point of intersection (CX) times endpoint of the chord to
the point of intersection (DX)

6. X
B
C
A
D

7. Note: secants intersect outside the circle
X
B
C
A
D

8. Note: secants intersect outside the circle
X
B
AX  BX  CX  DX
C
A
D

9. Note: secants intersect outside the circle
X
B
AX  BX  CX  DX
A
C
product of intercepts of intersecting secants
D

10. Note: secants intersect outside the circle
X
B
AX  BX  CX  DX
A
C
product of intercepts of intersecting secants
D
A
X
C
D

11. Note: secants intersect outside the circle
X
B
AX  BX  CX  DX
A
C
product of intercepts of intersecting secants
D
A
X
AX 2  CX  DX
C
D

12. Note: secants intersect outside the circle
X
B
AX  BX  CX  DX
A
C
product of intercepts of intersecting secants
D
A
X
AX 2  CX  DX
C
square of tangents  product of intercepts
D

13. Note: secants intersect outside the circle
X
B
AX  BX  CX  DX
A
C
product of intercepts of intersecting secants
D
Exercise 9G; 1ace, 2, 4, 6, 9a
A
X
AX 2  CX  DX
C
square of tangents  product of intercepts
D