100 Great Problems - Supplement

Ryan Hirst

A collection of theorems and constructions used in the proofs for [i]100 Great Problems[/i] : [url]http://www.geogebratube.org/material/show/id/73813[/url] which accompany the book by Heinrich Dorrie.


  • 1. The Secant Theorem
  • 2. Heron's Formula, Geometric Proof
  • 3. Medians and the Centroid
  • 4. Triangle Inscribed in a Circle
  • 5. Inscribe a circle in a triangle.
  • 6. A quadrilateral inscribed in a circle
  • 7. Tangents from point to circle
  • 8. The Diagonals of Trapezoid
  • 9. Inscribe a chord in a given arc of a circle
  • 10. Draw the sine squared of an angle
  • 11. Chordal: Points with equal tangents to two circles
  • 12. Draw a Chordal (Power Line)
  • 13. Similarity points
  • 14. Pole and Polar 1
  • 15. Pole and Polar 2
  • 16. Pappus' Theorem
  • 17. Pappus' Theorem 2
  • 18. Malfatti: Trig supplement
  • 19. Tangent circles, differential
  • 20. Appolonius, Differential Solution I
  • 21. Ruslan and Ludmila
  • 22. Cycloids and the Rolling Circle
  • 23. The Parabola: Focal angle of intersecting tangents
  • 24. Parabola from Four Tangents
  • 25. I am a rhombus
  • 26. Rhombus & Circle, 2

1.

The Secant Theorem

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

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