Artzt_2

The three Artzt parabolas of a triangle

For every triangle ABC there are three parabolas denoted by p_AB, p_BC, p_CA. Parabola p_AB passes through A, B and is tangent there to sides CA, CB respectively. Analogously are defined the other parabolas. File Artzt.html starts the discussion of basic properties of these curves. Here are some additional facts.
[1] Their axes are parallel to the medians of triangle ABC.
[2] Parabola p_AB is tangent to line DE at its middle, D, E being the middles of sides AC, BC. Analogous result for the other parabolas.
[3] Their focal points coincide with the vertices of the second Brocard triangle of ABC.
[4] Every pair of parabolas has an intersection point on the median which is parallel to the axis of the third parabola.
[5] The three intersection points A',B',C' of [4] define a triangle homothetic to DEF with respect to the centroid J of ABC and with a homothety ratio equal to 2/3.
[6] The tangent to parabola p_BC at A' is parallel to the axis of the other parabola through that point (parallel to median BE).
Artzt parabolas of 2nd kind are discussed in the file Artzt2.html .

[1,2] Follow from general facts on parabolas. Parabola p_BC for example is the member of the bittangent family U*V-k*W² = 0, passing through G. U, V denoting the equations of AB, AC and W that of line BC. [3] is discussed in the reference on the second Brocard triangle.
The precise location of points B', C' on the diagonals of trapezium FECB is discussed in the file ParabolaTrapezium.html .

The three Artzt parabolas of a triangle

See Also