A Tour of Triangle Geometry via the Geometer's Sketchpad

• 1. Reflections and isogonal lines
• isogonal conjugates
• reflections of P and P* have congruent circumcircles
• O and H are isogonal conjugates
• reflections of H lie on the circumcircle
• common pedal circle of O and H
• nine-point circle
• 2. Examples of isogonal conjugates
• Centroid and symmedian point (perspector of tangential triangle)
• Gergonne point and internal center of similitude of circumcircle and incircle
• Nagel point and external center of similtude of circumcircle and incircle
• isogonal conjugates of points on circumcircle
• isogonal conjugate of G
• perspector of tangential triangle
• perspector of triangle bounded by outer sides of similar rectangles erected on the sides
• two Lemoine circles
• center of conic tangent to sidelines at pedals of H
• Brocard circle (seven-point circle)
• OG : GN : NH = 2:1:3
• de Longchamps point L: reflection of H in O as radical center of triad of circles A(a), B(b), C(c)
• Schiffer point Sc: common point of the Euler lines of IBC, ICA, IAB, and ABC
• circumcenter of tangential triangle
• Euler infinity point and centroid of its cevian triangle
• 5. Line of reflections
• Simson line and line of reflections
• Simson lines of antipodal points intersect orthogonally on nine-point circle
• point with given line of reflections: reflections of line through H concur at a point on circumcircle
• reflections of Euler line concur at E on circumcircle
• circles APX, BPY, CPZ (X,Y,Z reflections of H in AP, BP, CP) intersect on circumcircle
• 6. Some conic constructions
• tangent at a point of  conic
• second intersection of line with conic
• center of conic
• de Longchamps point as the ``radical center'' of three ellipses
• Soddy circles
• 7. Rectangular hyperbolas
• Poncelet-Brianchon theorem (1822): A rectangular circum-hyperbola passes through the orthocenter and has center on the nine-point circle
• H(P) : rectangular hyperbola through P, center W(P), tangent at H
• reflections of tangent at H of rectangular circum-hyperbola through P
• asymptotes of rectangular hyperbola: regarded as infinite points, isogonal conjugates on circumcircle, antipodal. The hyperbola is the locus of isogonal conjugates of points on the circum-diameter.  H(P) as isogonal conjugate of OP*, fourth common point of H(P) and circumcircle: isogonal conjugate of infinite point of OP*
• isogonal conjugate of a line: circumconic
• 8. Three examples:
• Jerabek hyperbola: isogonal conjugate of the Euler line
• Kiepert hyperbola: isogonal conjugate of OK
• Feuerbach hyperbola: isogonal conjugate of OI
• 9. Reflection conjugates
• reflection conjugate r(P) (except for H and points on the circumcircle)
• r(P) = antipode of P in H(P),  P* and (rP)* inverse in circumcircle
• 10. Inscribed conics
• inscribed conic with prescribed foci P and P*
• inscribed ellipse with foci O and H, center N
• inscribed conic tangent to pedal circle
• construction of inscribed conic
• 11. Inscribed parabolas
• inscribed parabola: focus F on circumcircle, directrix = line of reflections of F
• inscribed parabola tangent to a given line
• inscribed parabola tangent to Euler line
• a cubic curve: locus of P for which the inscribed parabola tangent to OP touches it at P
• 12. Inverses in circumcircle
• inversive images of traces of a point P in circumcircle
• the case of G
• P on circumcircle: locus of  perspector - isogonal conjugate of nine-point circle
• P on Euler line: locus of perspector - conic through the traces of the isogonal conjugates of Kiepert and Jerabek centers
• 13.  Reflections of circumcevian traces
• Appendix A: reflections in altitudes