Venema Gerard A. Exploring Advanced Euclidean Geometry with GeoGebra

Mathematical Association of America, 2013 — 129 p. — (Classroom Resource Materials). — ISBN 9780883857847, 0883857847.This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry.
The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.Quick Review of Elementary Euclidean Geometry
Measurement and congruence
Angle addition
Triangles and triangle congruence conditions
Separation and continuity
The exterior angle theorem
Perpendicular lines and parallel lines
The Pythagorean theorem
Similar triangles
Quadrilaterals
Circles and inscribed angles
Area
The Elements of GeoGebra
Getting started: the GeoGebra toolbar
Simple constructions and the drag test
Measurement and calculation
Enhancing the sketch
The Classical Triangle Centers
Concurrent lines
Medians and the centroid
Altitudes and the orthocenter
Perpendicular bisectors and the circumcenter
The Euler line
Advanced Techniques in GeoGebra
User-defined tools
Check boxes
The Pythagorean theorem revisited
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 point and the Nagel point
Heron’s formula
The Medial and Orthic Triangles
The medial triangle
The orthic triangle
Cevian triangles
Pedal triangles
Quadrilaterals
Basic definitions
Convex and crossed quadrilaterals
Cyclic quadrilaterals
Diagonals
The Nine-Point Circle
The nine-point circle
The nine-point center
Feuerbach’s theorem
Ceva’s Theorem
Exploring Ceva’s theorem
Sensed ratios and ideal points
The standard form of Ceva’s theorem
The trigonometric form of Ceva’s theorem
The concurrence theorems
Isotomic and isogonal conjugates and the symmedian point
The Theorem of Menelaus
Duality
The theorem of Menelaus
Circles and Lines
The power of a point
The radical axis
The radical center
Applications of the Theorem of Menelaus
Tangent lines and angle bisectors
Desargues’ theorem
Pascal’s mystic hexagram
Brianchon’s theorem
Pappus’s theorem
Simson’s theorem
Ptolemy’s theorem
The butterfly theorem
Additional Topics in Triangle Geometry
Napoleon’s theorem and the Napoleon point
The Torricelli point
van Aubel’s theorem
Miquel’s theorem and Miquel points
The Fermat point
Morley’s theorem
Inversions in Circles
Inverting points
Inverting circles and lines
Othogonality
Angles and distances
The Poincar´e Disk
The Poincar´e disk model for hyperbolic geometry1
The hyperbolic straightedge
Common perpendiculars
The hyperbolic compass
Other hyperbolic tools
Triangle centers in hyperbolic geometryAbout the Author