[1] There is a conic c

[2] The parallel to B*C* from A meets c

[1] is a consequence of the inverse of Pascal's theorem applied to the hexagon ABB*A*CC*. Its opposite sides intersect at two real points (the side pairs (AB,A*C) and (A*B*,AC*)) and one point at infinity (sides (BB*,CC*)). Hence they are on a line, consequently there is a conic c

[2] is a consequence of the Pascal theorem. First note that A' is the center of the conic since it is the center of a rectangular paralellogram BB*CC* inscribed in c

There is an interesting third property of this figure related to the Euler circle of ABC and examined in the file PascalEuler.html .

Since the symmetric A'' of A with respect to A' is also on the ellipse, the previous remark gives a way to construct all the ellipses circumscribing a given parallelogram ABA''C. Simply take a fifth point D on the circumcircle of ABC and pass a conic through the five points A, B, A'', C and D. EucliDraw has the corresponding tool to do that.

Pascal.html

PascalEuler.html

Produced with EucliDraw© |