Conics

Illustrate Pascal's Mystic Hexagram Theorem for a Circle: The points 12,  23, 31 of the intersection of the three pairs of opposite sides 1'2 and 12', 2'3 and 23', 3'1 and 13' of a hexagon 12'31'23' inscribed in a circle lie on a line.

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Construct the conic passing through five given points.

 

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Construct the conic passing through four given points and tangent to a given line which contains one of the points.



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Given three points and two lines each containing one of the points,  construct the conic passing through the three points and tangent to the lines
 at the given points.

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Illustrate Brianchon's Theorem for a Circle: If a hexagon is circumscribed about a circle, the three joining pairs of opposite vertices are concurrent.

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Construct the conic tangent to five given lines.

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Construct the conic tangent to four given lines and passes through a point on  one of them.

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Construct the conic tangent to three given lines and passes through two points on two of them.

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Construct the ellipse tangent to two fixed circles and their external common tangents.