Autopolar

1. Autopolar or self-polar triangles

Triangle ABC is autopolar with respect to a circle (c) if each side-line of the triangle is the polar of the opposite lying vertex.

Given a circle (d) and a point C construct the polar line c of C with respect to d. Then take any point A on this line and define the polar (a) of A with respect to c. Line a intersects c at a point B. Triangle ABC is an Autopolar (or self-polar or conjugate or polar) with respect to circle (d).
Circle (d) is called the conjugate circle of the triangle. Below are discussed some elementary properties of autopolar triangles.

[1] Every autopolar triangle t = ABC is defined by the procedure described above.
[2] All autopolar triangles have a unique vertex inside the circle (d) and two outside.
[3] They are obtuse and have orthocenter the center O of circle (d).
[4] Their circumcenters Q are at distance |CO|/2 from the line (c), opposite to the obtuse angle.
[5] The inversion F with respect to d inverts the circumcircle (e) to the Euler circle (f) of t.
[6] Circle (d) belongs to the coaxal circle bundle I(e,f) of intersecting type generated by the circumcircle (e) and the Euler circle (f) of triangle t.
[7] The previous property together with the property (3) of O, uniquely define the circle (d) from the data of triangle t.
[8] The tangents AA₁, AA₂ from A to (d) have contact points along line BC and are harmonic conjugate to B, C. Analogous property holds for the tangents to (d) from B.
[9] The previous properties imply that every obtuse triangle is autopolar with respect to a uniquely determined circle which coincides with the one defining an inversion interchanging the circumcircle and the Euler circle.

Property (5) follows from the fact that the products of lengths OA*OF = OB*OE = OG*OC = r² . Hence the inversion with center O and radius r maps the circumcircle to the Euler circle. It remains to show that circle (d) has the radius r. But this follows from the definition of the polar and the basic facts on harmonic tetrads. Indeed, extend GC to define its intersection points X, Y with the circle (d). Since c is the polar of C, (X,Y,C,G) = -1 is a harmonic tetrad and this implies OX² = OC*OG = r². (5) and (6) are consequences of this property. The other properties are easy to show.

2. Another characteristic construction

Another way to generate self-polar triangles is the following: Consider a triangle ABC and a point P on its circumcircle. The cevian triangle A'B'C' of P i.e. the triangle formed by joining P to the vertices and intersecting the opposite sides of ABC is a self-polar triangle.
This follows from the basic method by which we construct polars (see Polar2.html ). According to this, considering the quadrilateral ABCP and viewing {A',C'} as intersections of its opposite sides we see that B'C' is the polar of A' and B'A' is the polar of C'. From this follows that A'C' is the polar of B'.

Some additional properties of this configuration are the following:
[1] The sides of the self-polar triangle A'B'C' pass through the vertices of the tangential triangle A''B''C''.
[2] Pairs of sides of A'B'C' and A''B''C'' intersect on the tangent at P (namely (A'B',A''B''),(B'C',B''C''),(C'A',C''A'')).
[3] Triangles A'B'C' and A''B''C'' are perspective from a point P' which is the tripole of the tangent at P with respect to A'B'C'.
[4] P' lies on the Lemoine axis of triangle ABC and is also the tripole of of the tangent at P with respect to the tangential triangle A''B''C''.
The properties listed here concern a special case of a more general configuration discussed in Autopolar2.html .
Remark-1 The definition and properties of harmonic tetrads of points on a line are discussed in Harmonic.html . The definition and properties of polars are discussed in the file Polar.html . A discussion on the inversion interchanging two given intersecting circles can be found in MidCircles.html .
Remark-2 Autopolar triangles appear naturally in products G*F of perspectivities which are commutative (F*G = G*F), see the file FourPointsCyclic.html for a brief discussion.
Remark-3 Autopolar triangles are defined also with respect to a conic. See Autopolar2.html for the more general definition as well as the generalization of properties of section-2.

References

Deltheil, R & Caire, D. Geometrie et Complements Paris, Editions Gabay, 1989, p. 135.

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1. Autopolar or self-polar triangles

2. Another characteristic construction

See Also

References