AllParabolasCircumscribed

1. All parabolas circumscribed about a triangle

Problem: To construct all parabolas passing through three given points A, B, C.

The solution can be obtained in a very simple and structural way by using the property of the anticomplementary triangle A'B'C' related to the axis of the parabola discussed in AnticomplementaryAndCircumparabola.html .
There it was shown that the parallels to the axis of a circumparabola of ABC from the vertices of the anticomplementary triangle A'B'C' meet the opposite sides of A'B'C' on the parabola.
Thus, taking a pivoting point D on the circumcircle and defining the axis direction through line OD (O the circumcenter of ABC), we obtain all circumparabolas of ABC by determining three additional points {A'',B'',C''} through which the parabola should pass.
{A'',B'',C''} are respectively the intersection points of the parallels to line OD from the vertices {A',B',C'} with the opposite sides of this triangle.

Having six points on the parabola we can draw it with the tool drawing a conic through five points (selecting five of them). For another way to generate a circumparabola see the file CircumparabolaGeneration.html .
Note that there are three particular directions of OD for which the parabola degenerates to a pair of parallel lines. These occur when OD is parallel to the sides of the triangle. For example in the case OD tends to be parallel to AB, points {A,B,C''} and {A'',B'',C} become corrspondingly collinear, the two lines supporting them being line AB and its parallel from C.

Remark-1 The figure shows also the locus of the corresponding perspector of the circumparabola. It is the inner Steiner ellipse of triangle ABC. By the way, I mention here the structural role of the inner Steiner ellipse of a triangle. For perspectors on this ellipse the correpsonding conic is a parabola. Perspectors located inside the ellipse produce ellipses and perspectors located outside this ellipse correspond to hyperbolas.

Remark-2 There are various additional properties here like, for example, the tangency of the parabola to the line at infinity, which in turn is the trilinear polar of G, the centroid of the triangle. The generalization of all these properties are discussed in a more general setting in the file CircumconicsTangentToLine.html .

2. Parabolas trhough four points

Given four points {A,B,C,D} in general position we can use the preceding construction to explore the existence of parabolas passing through these points. For this consider the set S of all parabolas through {A,B,C} and determine the locations of D for which there is a parabola in S passing through D.

It is simple to see that every parabola passing through {A,B,C} has no points inside the angular domains indicated above. Thus, (i) for every point D₀ lying in these domains there is no parabola passing through {A,B,C,D₀}, (ii) for every point D₂ in the open complement of these domains there are always two parabolas passing through {A,B,C,D₂}.
Concerning the existence of these parabolas one can apply the general theory of conics passing through four points and tangent to a given line (see FourPtsAndTangent.html ). The line in the present case is the line at infinity.
Remark The condition on the existence of parabolas, through four points stated above using the indicated domains, is equivalent to the existence of a convex quadrangle with vertices {A,B,C,D}.

3. Parabolas trhough four points, their axes

Given four points {A,B,C,D} satisfying the convexity property of the previous section (remark), there are two real parabolas through these points. Their axes can be determined using the Desargues involution theorem (see DesarguesInvolution.html ).

In fact all conics passing through these four points induce, through their intersection points, on any line L a homographic involution (see InvolutionBasic.html ). Selecting the line L to be the line at infinity, the two parabolas correspond to the fixed points of this involution, which translates to the directions of the axes of the parabolas.
Thus, identifying the line at infinity with the pencil O* of lines through an arbitrary point O, we can construct the two directions of the parabolas as fixed points of an involution on O*.
[1] The involution is defined by drawing an arbitrary line L and considering its intersection points with the lines of O*.
[2] More precisely, given the four points {A,B,C,D} consider the pairs of points on L (A',C'), (B',D') obtained by intersecting L with parallels respectively to {FA,FC,EB,ED}.
[3] The two pairs (A',C'), (B',D') uniquely define the involution on L, which can be represented by the circles of the circle-bundle generated by the circles on diameters A'C' and B'D' (see Involution.html ).
[4] The fixed points of this involution are the limit points M, N of this circle bunde. They can be constructed as intersections of L with an arbitrary circle (c) orthogonal to the circles with diameters A'C' and B'D'.
[5] The axes of the parabolas are parallel to lines OM and ON.

4. The case of trapezia

When the four given points {A,B,C,D} are vertices of a trapezium, then there is only one non-degenerate parabola, whose construction is considered in ParabolaCircumscribingTrapezium.html . The other one is the degenerate parabola represented by the union of the two lines carying the parallel sides of the trapezium.

1. All parabolas circumscribed about a triangle

2. Parabolas trhough four points

3. Parabolas trhough four points, their axes

4. The case of trapezia

See Also