What is the Multiplication Rule for AD and CD in a Circle with a Diameter BE?

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SUMMARY

The discussion focuses on the Multiplication Rule for segments AD and CD in a circle with diameter BE. It establishes that for a circle centered at point P with radius PA equal to PB, the relationship AD·CD = BD·DE holds true. Specifically, the calculation shows that AD·CD equals 7, derived from the segments where BD is 1 and DE is 7 (8 - 1). This geometric principle is reinforced by the fact that the angle at the center is twice the angle at the circumference.

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  • Understanding of circle geometry and properties
  • Familiarity with the concept of diameters and radii
  • Knowledge of angle relationships in circles
  • Basic algebra for manipulating geometric equations
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  • Study the properties of circles, focusing on the relationship between angles at the center and circumference
  • Explore geometric proofs involving the Multiplication Rule in circle theorems
  • Learn about segment relationships in circles, particularly with diameters
  • Investigate other geometric rules that apply to circles, such as the Power of a Point theorem
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This discussion is beneficial for mathematics students, educators teaching geometry, and anyone interested in advanced circle theorems and their applications.

Albert1
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[sp]The circle centred at $P$ with radius $PA = PB$ passes through $C$ (angle at centre = twice angle at circumference). Extend $BP$ to form a diameter $BE$ of the circle. Then $AD\cdot CD = BD\cdot DE = 1\cdot (8-1) = 7$.[/sp]
 
Opalg said:
[sp]The circle centred at $P$ with radius $PA = PB$ passes through $C$ (angle at centre = twice angle at circumference). Extend $BP$ to form a diameter $BE$ of the circle. Then $AD\cdot CD = BD\cdot DE = 1\cdot (8-1) = 7$.[/sp]
very good !
 

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