TrianglesCircumscribingParabolas

Triangles Circumscribing Parabolas

Here are some basic properties of a triangle t=ABC, having its sides tangent to a parabola (c).
1) The triangles FBD, FBG and FCA are similar.
2) The focus F lies on the circumcircle of the triangle.
3) The projections of the focus F on the sides of the triangle lie on the tangent (b) at the vertex V of the parabola. (b) is the Simson line of the focus F w.r. to the triangle ABC.
4) The directrix (a) of the parabola passes through the orthocenter H of the triangle.

1) Follows from the discussion on the triangle BDG (see ParabolaChords.html ). There we saw that s₁=BFG and s₂ = BFD are similar. Besides we saw also that, J, K, D being the projections of F on the sides of ABC, DJF is also similar to s₁, s₂ and from the cyclicity of quadrilaterals KFJC and AFKD, AFC is similar to s₃ = DFJ.
2) From (1) and the cyclicity of AFCB, follows that F is on the circumcircle of ABC.
3) Then the projections of F on the sides, define the Simson line of F, which, by the basic properties of the parabola coincides with the tangent at the vertex V.
4) The directrix (a), being the "double" (Steiner line) of the line (b) (GK = KF, IJ=JF, etc.) passes through the orthocenter (a general property of Steiner lines).

Consider the two tangents fixed, say BG and BD, and the third AC moving (E moving on c). The triangle AFC remains similar to itself as E moves on c. Thus, we can generate the parabola by the following recipe:
Consider a triangle AFC, moving so that F is fixed, A glides on a fixed line BG, C glides on a line BD and the triangle remains similar to itself. Then its side AC, opposite to the fixed point F, envelopes a parabola (c) with focus F and having the fixed lines BG, BD as tangents.

Triangles Circumscribing Parabolas

See Also