SimsonGeneral

Generalized Simson Line

Following Poncelet draw from a point J of the circumcircle of triangle ABC lines at an angle (phi) to the sides of the triangle. The intersection-points of these lines with the respective sides of the triangle are on a line depending on the position of J. For (phi) equal to (pi)/2 this is the well known theorem of Wallace-Simson (see Simson.html ).

Because of the way they are defined, the quadrangles JIGA, JIBH and JGCH are cyclic. Compare angles: ang(JGI) = ang(JAI) and ang(JGH) = ang(JCH). But A, B, C, J concyclic => ang(JAI) = ang(JCB) = ang(JCH). Hence G, I and H are on a line.
By Miquel's theorem (see Miquel_Point.html ) the centers of the circles N, M, L and O, together with J are on a circle and the triangle JKL is isosceles fixed, independent of the position of J on the circumcircle of ABC. The angles (psi) at its basis are supplementary to the angle (phi): (phi) + (psi) = (pi)/2. The other intersection point P of the circumcircle with the Miquel circle (P symmetric of J with respect to KL) has Simson line [QR] parallel to the generalized Simson line [HI].
The envelope of Simson lines of ABC is a well known deltoid. The envelope of the generalized Simson lines [HI] is a similar deltoid. Both deltoids can be viewed in SimsonGeneral2.html .

References
H. A. Converse, On a System of Hypocycloids of Class Three Inscribed to a Given 3-Line, and Some Curves Connected with it The Annals of Mathematics, Second Series, Vol. 5, No. 3, (Apr., 1904), pp. 105-139

Generalized Simson Line

See Also