AnticomplementaryAndCircumparabola

Anticomplementary and circumparabola

Let the vertices of our basic triangle, points {a,b,c}, be on the parabola 4ky-x²=0 (k is the focus-vertex distance). Consider also points {a'=b+c-a, b'=c+a-b, c'=a+b-c} which are the vertices of the anticomplementary triangle.
The parallels from {a',b',c'} to the axis of the parabola meet the opposite sides of the anticomplementary triangle on points of the parabola.

In fact, the parallel line L₁ from a'=b+c-a, is the line having constant x-coordinate and equal to (b₁+c₁-a₁). The line L₂ through {b',c'} in parametric form is b'+t(c'-b) and meets L₁ at a point such that b'₁+t(c'₁-b₁)=b₁+c₁-a₁. From which follows immediately that
t=(b₁-a₁)/(b₁-c₁).
Thus, the y-coordinate of the intersection point is b'₂+t(c'₂-b'₂)=a₂+(2t-1)(b₂-c₂) , which replacing y₂=x₁²/(4k) reduces to (1/(4k))(b₁+c₁-a₁)², thus proving the claim.

Remark-1 It follows that for every tripple of non-collinear points {A,B,C} and every direction L, there is conic passing through the six points {A,B,C,A'',B'',C''}, where the three last points are the intersections of the parallels to L from the vertices and the opposite sides to these vertices of the anticomplementary triangle.
Further, for all directions not coinciding with the sides of ABC the conic is a parabola whereas for directions coinciding with a side of the triangle the corresponding conic degenerates to a set of two parallel lines.
Remark-2 Non degnerate parabolas touch the line at infinity at their "point" corresponding to the direction of the axis. This is used in the reference given below.

The remarks have a nice consequence for parabolas circumscribing a triangle. See the file CircumconicsTangentToLine.html . Another approach of the same subject is to be found in AllParabolasCircumscribed.html .

Anticomplementary and circumparabola

See Also