General Linear Transformation: Free Points (matrix columns: Bx, By)
The lines at the center points of the sides are obtained by a general linear transformation.
The two column vectors for its matrix are shown in light blue color.
In this general case, the resulting lines do not always meet at a common point (green triangle).
No "circumcenter theorem" can be thus established for this general case.
The matrix for this general linear transformation can be written as a function of two column vectors: (Bx, By)
Bx1 By1
Bx2 By2
You can try to find out which positions of the blue points produce a common intersection (green triangle of zero size).
Restricted Linear Transformation: Linked Points
We can restrict the liberty of the vector columns in te following way: A sliding point (red) is put at the diagonal with negative slope (red line). The column vectors are marked by sliding points on the vertical (first vector, in light green color) and horizontal (second vector, in light blue) lines which coincide at this red slider.
In this case, the three "mediatrices" meet at a common point: we have a "theorem".
The matrix for this special transformation can be written as a function of the sliding point (D,-D):
D By1
Bx2 -D
We can also look at the grey quadrilateral and recognise that when it is arrow-shaped (concave) then the circumConic is an ellypse, while other shapes (convex or crossed) render a hyperbola.
When the quadrilateral degenerates into a triangle we have a limiting situation. In this case, the circumConic is a parabola, and the matrix has the following form:
D -aD
D/a -D
The determinat of this matrix is obviously zero, which corresponds to a degenerated linear transformation. It projects the whole plane into a line, whose direction coincides with that of the parabola´s axis.