Ilustrate this theorem: Let ABC be a given triangle and O an arbitrary
point of the plane. Draw AO, BO, CO to meet BC, CA, AB in L, M, N respectively, and then
draw MN, NL, LM to meet BC, CA, AB in U,V,W respectively. Then U,V,W are collinear. The
straight line passing through U,V,W is called the |

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Suppose the triangle ABC and a straight line in m the plane are given.
Construct the unique point O having m as polar. This point is called the pole
of the line m. |

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Illustrate this: Let ABCDE be an arbitrary pentagon, F the point of intersection of the nonadjacent sides AB and CD, M the point of intersection of the diagonal AD with the line EF. Then the point of interestion P with the side AE with the line BM, the point of intersection Q of the side DE with the line CM, and the point of intersection R of the side BC with the diagonal AD all lie on one line. |

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Illustrate Pappus Theorem: If the vertices of a hexagon
fall alternatively on two lines, the intersections of opposite sides are collinear. |

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Illustrate Theorem of Desargues: If two triangles have a
center of perspective, they have an axis of perspective. |

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Illustrate Theorem on Doubly Perspective Triangles: Two
doubly perspective triangles are in fact triply perspective. |

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Illustrate Theorem on Triply Perspective Triangles: Two
triply perspective triangles are in fact quadruply perspective. |

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Illustrate this theorem: Let O be a point on the plane of a triangle ABC,
and let A_{1}, B_{1}, C_{1} be the points of intersection of the
lines AO, BO, CO with the sides of the triangle opposite A,B,C. If A_{2}, B_{2},
C_{2} are on B_{1}C_{1}, C_{1}A_{1}, A_{1}B_{1}
such that the three lines A_{1}A_{2}, B_{1}B_{2}, C_{1}C_{2}
are concurrent, then AA_{2}, BB_{2}, CC_{2} are concurrent. |

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