PascalEuler

A remarkable control point on the Euler circle

Consider a triangle ABC and reflect it on a line a' passing through the middle A' of its side BC to the triangle A*B*C*.
[1] There is a conic c_a passing through all six vertices of triangles ABC, A*B*C*.
[2] The parallel to B*C* from A meets c_a a second time at a point D lying on the circumcircle c of ABC.
[3] The intersection point D of the conic c_a and c, the centroid G of ABC and the intersection point D*, other than A' of the Euler circle of ABC and B*C* are collinear. Further the ratio DG/GD*=2.
[1] and [2] were discussed in PascalImmortel.html . Here we discuss the third property.

The third property is a trivial consequence of the considerations in the cited reference. There we proved that AD is parallel to B*C*. Since OA and FA' are also parallel (see Euler.html ) the isoscelli OAD and FA'D* are similar, and the same is true with triangles FGD* and GOD. Hence the intersection point of DD* with OF coincides with the centroid G of ABC and the ratio DG/GD*=OD/FD*=2.
From this property follows that the shape of the conic c_a is controlled from the position of point D* on the Euler circle of ABC. One could start from this point, which defines the reflexion axis a' as the bisector of angle BA'B* and point D on line D*G.
Since all conics c_a pass also from the symmetric H of A with respect to A' the subject has tight relations with the subject of all conics circumscribing a parallelogram (like ABHC).

A remarkable control point on the Euler circle

See Also