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.

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 D

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.

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.

CircumconicsTangentToLine.html

CircumparabolaGeneration.html

DesarguesInvolution.html

Involution.html

InvolutionBasic.html

ParabolaCircumscribingTrapezium.html

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