A Tour of Triangle Geometry via the Geometer's Sketchpad

Paul Yiu
Department of Mathematics
Florida Atlantic University

37th Annual Meeting of the Florida Section of MAA
University of Central Florida, Orlando,
February 25--26, 2004

• 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
  
• 3. The symmedian point K: 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)
  
• 4. The Euler line: 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
  
• Appendix A: reflections in altitudes
• Appendix B:  barycentric coordinates