Geometric Constructions 9

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.

Construct
the conic passing through five given points.

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

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

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

Construct
the conic tangent to five given lines.

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

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