Please see:
http://mathworld.wolfram.com/ChaslessPolarTriangleTheorem.html
http://forumgeom.fau.edu/FG2004volume4/FG200427.pdf
I proposed problem construction of a rectangular hyperbolar and a inscribed parabola as follows:
Let ABC be a triangle, let a circle with radii R, A0B0C0 is the triangle bounded by the polar of A,B,C to the circle. A1 is the intersection of the polar of A and BC; define B1,C1 are cyclically. By Chasless theorem we have A1,B1,C1 are collinear. We call A1B1C1 is the Chasless line. Now by Desargues' theorem we have: ABC and A0B0C0 are perpective. Denote the perpector is P. Show that:
1-P lie on a rectangular hyperbola through center of the circle when R changed. (I mean six point A,B,C, the orthocenter, P and center of the circle lie on a hyperbola with any R).
2-The Chasless line are tangent with a inscribed parabola when R changed (or center of the circle be moved on the line through the orthocenter of the triangle ABC and original location of center of the circle.)
http://mathworld.wolfram.com/ChaslessPolarTriangleTheorem.html
http://forumgeom.fau.edu/FG2004volume4/FG200427.pdf
I proposed problem construction of a rectangular hyperbolar and a inscribed parabola as follows:
Let ABC be a triangle, let a circle with radii R, A0B0C0 is the triangle bounded by the polar of A,B,C to the circle. A1 is the intersection of the polar of A and BC; define B1,C1 are cyclically. By Chasless theorem we have A1,B1,C1 are collinear. We call A1B1C1 is the Chasless line. Now by Desargues' theorem we have: ABC and A0B0C0 are perpective. Denote the perpector is P. Show that:
1-P lie on a rectangular hyperbola through center of the circle when R changed. (I mean six point A,B,C, the orthocenter, P and center of the circle lie on a hyperbola with any R).
2-The Chasless line are tangent with a inscribed parabola when R changed (or center of the circle be moved on the line through the orthocenter of the triangle ABC and original location of center of the circle.)
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