ParabolaTangentEquilateral

1. Parabola inscribed in an equilateral triangle

Some general properties of parabolas inscribed to triangles are discussed in ParabolaInscribed.html . Here is the figure of the special case of an equilateral triangle ABC to the sides of which the parabola is tangent. I do not repeat the statements and proofs of coincidences and collinearities suggested by the figure. They all are reviewed in the aforementioned reference.

Here I only notice the fact that the perspector P of the parabola lying on the circumcircle of ABC is also the focus of the parabola.

The fact that P is the focus of the parabola follows from the equality of segments PC'=C'C₀ and PB'=B'B₀. Here {B₀, C₀} are the projections of the corresponding contact points {B',C'} on tr(P), which is also the polar of P with respect to the parabola. To see this equality notice that CC'' (A''B''C'' is the anticomplementary of ABC) has AC' as medial line hence the angles CC'A and AC'C'' are equal. Also tr(P) is the Steiner line of point P with respect to ABC and is orthogonal to the direction of the parallels (see IncircleTangents.html ) hence the symmetrics of P on the sides of ABC are on tr(P) (see SteinerLine.html ). Thus the symmetric of P on side AB is on the lines tr(P) and CC'' hence coincides with their intersection C₀. Analogous argument shows that B'B₀=B'P thereby implying that P is the focus and tr(P) the directrix of the parabola.

2. Equilaterals tangent to a parabola

The above figure suggests that there are many equilateral triangles with sides tangent to a given parabola. In fact, selecting three directions {AA',BB',CC'} through the focus F such that their mutual angles are all equal and taking tangents at their intersection points {A,B,C} with the parabola we form a tangential triangle as required.

The proof follows from Brianchon's theorem and the passing of the circumcircle of A'B'C' through the focus of the parabola (see InconicsTangents.html and ParabolaChords.html ). The theorem of Brianchon implies that lines {FA,FB,FC} pass through the vertices of the tangential triangle. The property of the circumcircle implies that the angles at {A',B',C'} are each π/3 and the triangle is equilateral. Then the Steiner line of F is the directrix and the orthogonal to this line is parallel to the axis of the parabola.

1. Parabola inscribed in an equilateral triangle

2. Equilaterals tangent to a parabola

See Also