[2] Generate similar triangles to the previous three by taking points {B

[3] Set AB/AC=k. Then AB

[4] This implies that the right angled triangles {AB

[5] By the generalization of Thales theorem (see Thales_General.html ) lines B

[6] All circumcircles of triangles OB

[7] Line AD is also tangent to the parabola. This is seen by letting B

[8] When B

[9] The parabola is easily constructible as a conic tangent to the five lines {OB,OC,A

[10] There are also two other, easy to construct positions of B

[11] Line OF is harmonic conjugate of OA with respect to lines {OB,OC}. This follows from O being on the polar of A (see [8]). Thus, by the reciprocity of polars, the polar of O passes also from A. This implies that {OB,OC,OA,OF} is a harmonic bundle.

[12] A consequence of the previous is that the tangent to circle (c) at A passes also through the intersection point I of BC and OF.

[13] Note that, according to general properties of triangles with sides tangent to a parabola, the tangent of the parabola at its vertex coincides with the Simson line of such a triangle with respect to the focus. Thus, this line in this case can be constructed by projecting F on line OB, OC.

[14] According to [11] F could be constructed by intersecting the circumcircle of OA

Similarly_Rotating.html

Thales_General.html

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