The solution can be obtained by using the following property:

Consider a conic (c) circumscribing a triangle t=ABC and tangent to a line (e) at a point D. Let A*, B*, C* be the intersection points of the tangent with the sides of t. Let further E be the intersection point of AA* and BB*, F the intersection point of the conic with CC*. Then points D, E, F are on a line (see ThreeCollinearPts2.html ).

The previous property can be used to determine all conics (c) passing through three points A, B, C, and tangent to a fixed line e. In fact, E is determined by the given data and D can be taken arbitrary, F being then determined by the intersection of the lines DE (variable and turning about E) and the fixed line CC*. The five points {A,B,C,D,F} determine a conic uniquely. The conics obtained in this way include the singular conics consisting of the pairs of intersecting lines: (BB*, AC), (AA*, BC), (CC*,AB). A particular case occurs when (e) is the line at infinity. Then the conic is a parabola (tangent to the line at infinity) and CC* is parallel to AB, AA* parallel to BC etc. See the file AllParabolasCircumscribed.html for the details.

Note that the family of conics contains exactly three parabolas, obtained when D is such that one of the following pairs of lines consists of parallels: (ED, E'E''), (E'D, E''E), (E''D,EE'). Thus, there are three parabolas passing through three given points and tangent to a given line. Their axes are parallel to lines EE', E'E'' and E''E.

Note that this construction can be done by considering any one of the intersection points E, E', E'' of lines AA*, BB*, CC*, giving, in total, three additional points F, F', F'' on the conic. The resulting families of conics though coincide. To view the various conics circumscribed about the triangle ABC catch (CTRL+2) and move the point D.

The case of line (e) separating the three points can be handled similarly and the resulting family of conics consists only of hyperbolas (and pairs of intersecting lines), see the corresponding illustration in the file AllConicsCircumscribed2.html .

Another approach of the same subject can be found in the file CircumconicsTangentToLine.html .

AllParabolasCircumscribed.html

CircumconicsTangentToLine.html

ThreeCollinearPts2.html

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