MTH 243

INTRODUCTION TO GEOMETRY

Advanced Euclidean Geometry. Topics may include the historical foundations of geometry, a review of basic Euclidean geometry, collinearity, concurrency, cross ratio, inversion, poles and polars, Hilbert's Axioms of geometry, and an introduction to non-Euclidean geometry. Dynamic geometry software will be used to illustrate theorems.


Course Outline

  • Elementary Euclidean Geometry (Review)
    • The Great River Civilizations
    • Measurement and congruence
    • Transformations
      • Isometries
        • Translations
        • Rotations
        • Reflections
      • Homotheties
        • Dilations
    • Angle Addition
    • Triangles and triangle congruence
    • Separation and continuity
    • The exterior angle theorem
    • Perpendicular and parallel lines
    • The Pythagorean theorem
    • Similar triangles
    • Quadrilaterals
    • Circles and inscribed angles
    • Area
  • Classical Triangle Centers
    • Concurrent lines
    • Medians and the centroid
    • Altitudes and the orthocenter
    • Perpendicular bisectors and the circumcenter
    • Angle bisectors and the incenter
    • The Euler line
  • Circumscribed, Inscribed, and Escribed Circles
    • The circumscribed circle and the circumcenter
    • The inscribed circle and the incenter
    • The escribed circles and the excenters
    • The Gergonne and Nagel points
    • Heron's Formula
  • Medial and Orthic Triangles
    • The medial triangle
    • The orthic triangle
    • Cevian triangles
    • Pedal triangles
  • Quadrilaterals
    • Convex and closed quadrilaterals
    • Cyclic Quadrilaterals
    • Diagonals and Brahmagupta's theorem
  • The Nine-Point Circle
    • The nine-point circle
    • The nine-point center
    • Feuerbach's theorem
  • Ceva's Theorem
    • Sensed ratios and ideal points
    • The theorem of Ceva
    • Concurrence theorems
    • Isotonic and Isogonal conjugates
    • The symmedian point
  • Menelaus' Theorem
    • Duality
    • The theorem of Menelaus
  • Circles and Lines
    • The power of a point
    • The radical axis
    • The radical center
  • Applications of the Theorems of Ceva and Menelaus
    • Tangent lines and angle bisectors
    • Desargues' theorem and projective geometry
    • Pascal's theorems
    • Brianchon's theorem
    • Pappus' theorem
    • Simson's theorem
    • Ptolemy's theorem
    • the butterfly theorem
    • Harmonic division
    • The golden section
  • Additional Topics in Triangle Geometry
    • Napoleon's theorem
    • The Torricelli point
    • vanAubel's theorem
    • Miquel's Theorem
    • The Fermat point
    • Morley's theorem
    • The Lemoine point
  • Inversions in Circles
    • Inverting points
    • Inverting circles and lines
    • Orthogonality and coaxial circles
    • Angles and distances
  • Solid Geometry
    • Polyhedra
    • The Platonic solids
    • More theorems of Euler
    • Archimedian Solids and duals
  • Non Euclidean Geometry
    • Spherical and elliptical geometry
    • Hyperbolic geometry (Poincarè model)
Contact R.L. Pryor