## Pascal: Necessary and Sufficient

The applet below illustrates the following construction. Given three points P, Q, R and a circle C(O). Choose a point A on the circle and extend AP to meet the circle in B'. Extend B'Q to meet the circle in C. Extend CR to meet the circle in A'. Extend A'P to meet the circle in B. Extend BQ to meet the circle in C' and, finally, extend C'R to meet the circle in, say, X. Under what conditions will X coincide with A?

What if applet does not run? |

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Copyright © 1996-2018 Alexander BogomolnyThe question is surely related to Pascal's theorem: from Pascal's theorem it follows that if

What if applet does not run? |

So, assume P, Q, R are collinear but

(This argument is a very slight modification of the one from the Diameters and Chords article.)

### Pascal and Brianchon Theorems

- Pascal's Theorem
- Pascal in Ellipse
- Pascal's Theorem, Homogeneous Coordinates
- Projective Proof of Pascal's Theorem
- Pascal Lines: Steiner and Kirkman Theorems
- Brianchon's theorem
- Brianchon in Ellipse
- The Mirror Property of Altitudes via Pascal's Hexagram
- Pappus' Theorem
- Pencils of Cubics
- Three Tangents, Three Chords in Ellipse
- MacLaurin's Construction of Conics
- Pascal in a Cyclic Quadrilateral
- Parallel Chords
- Parallel Chords in Ellipse
- Construction of Conics from Pascal's Theorem
- Pascal: Necessary and Sufficient
- Diameters and Chords
- Chasing Angles in Pascal's Hexagon
- Two Triangles Inscribed in a Conic
- Two Triangles Inscribed in a Conic - with Solution
- Two Pascals Merge into One
- Surprise: Right Angle in Circle

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Copyright © 1996-2018 Alexander Bogomolny69973990