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

Circle (d) is called the

[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 AA

[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 = r

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

[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

The properties listed here concern a special case of a more general configuration discussed in Autopolar2.html .

Euler.html

FourPointsCyclic.html

Harmonic.html

HyperbolaRectangular.html

MidCircles.html

NinePointsConic.html

OrthoRectangular.html

RectHypeImpossible.html

RectHyperbolasTangent4Lines.html

RectHyperbolaTriaInscribed.html

RectHypeThroughFourPts.html

Tangent4Lines.html

Produced with EucliDraw© |