TrianglesGivenPivot

1. A conic construction

Let c be a conic and A a fixed point outside it. Draw from A lines in varying directions intersecting the conic at points {B,B'}. Let also points {D,C} be the intersections with c of the polar of A with respect to c.
The intersection point I of line BD and the tangent t_B' to c at B' lies on a conic c'.
Conic c' is bitangent to c at points D and C (see BitangentConics.html ).
The proof reduces to a trivial property of equilateral hyperbolas (see HyperbolaRelatedToEquilateral.html ).
This is done by defining a homography which maps the conic c to a circle c₀ and the point A to a point at infinity.
Such a homography is easily constructed by coordinates-correspondence with respect to self polar triangles, and taking one of the sides of the autopolar triangle to pass through A.

2. Triangles with given pivot

The previous construction can be used to define a triangle inscribed in a conic c for which the conic has a given pivot. In fact, assume F is a given point outside a conic and we want to construct a triangle inscribed in the conic c and having the given point F as the pivot of the conic with respect to this triangle. For this it suffices to fix a point D on the conic. There is then a unique triangle DD'D'' inscribed in the conic, which then has for pivot the given point F.In fact, let A be the intersection point of the polar p_F of F with the tangent t_D of c at D.
Draw from A lines intersecting the conic at points BB' and construct the locus-conic (d) of the previous section. The intersection points {D₁,D₂} of this conic with the polar p_F of F determine through lines {DD₁,
DD₂} the other vertices of the desired triangle.
See the file TrianglesGivenPivotCanonical.html for a special construction along these lines.

3. Remarks

The locus of points I is invariant with respect to the harmonic perspectivity f with axis CD and center A. The polar of I passes from B'. Similarly the polar of I', constructed as the intersection of B'D and the tangent at B (thus belonging to the locus), passes through B. If follows easily that line II' passes through A and points I, I' are conjugate with respect to the harmonic perspectivity f.

4. Circle and its center

The case of circle and its center is easy to handle. Besides it delivers another way to handle the general case by mapping the general case through a homography to this particular one.

In fact, given a point A on the circle and its tangent there, consider another point B variable on the circle and draw from there the parallel BC to the tangent, C being the other intersection point of this parallel with the circle. The tangent t_C at C becomes parallel to AB exactly when the triangle ABC is equilateral.This simple fact identifies the equilaterals inscribed in the circle as the unique triangles ABC for which the center is the pivot of the circle with respect to ABC.

1. A conic construction

2. Triangles with given pivot

3. Remarks

4. Circle and its center

See Also