Objectives and competences
The objective of this course is to acquaint students with the basic concepts and results in plane and solid geometry. Two key approaches are presented and compared: the classical method and the approach through transformations. Students are directed to illustrate, apply and analyze these results using computer programs for dynamic geometry.
Content (Syllabus outline)
Triangle. Ceva's theorem. Basic triangle centers. Euler line, Nine-point circle. Simson line, Ptolemy's theorem.
Circle. Power of a point with respect to a circle, radical axis, radical center. Euler's theorem.
Quadrangles. Varignon's theorem. Cyclic quadrangles. Napoleon triangles. Area. Brahmagupta's formula. Heron's formula.
Menelaus' theorem and applications.
Transformations: isometries (translation, rotation, reflection), clasiffication theorem. Dilatation. Inversion.
Solid. Prism, Cylinder, Cone, Sphere. Volume. Angles in solids. Solid angles. Euler polyhedral formula. Platonic solids.
Learning and teaching methods
• Lectures
• Theoretical exercises
• Excersises in computer room.
• Teaching and learning are done through the didactic use of ICT
• Individual work
Intended learning outcomes - knowledge and understanding
On completion of this course, the student will be able to:
• Demonstrate knowledge and deep understanding of the results of plane and solid geometry.
• Provide proofs of geometric claims using classical methods and transformations.
• Conduct geometric constructions (using classical tools and using computer geometry tools).
• Confidently use one of the computer programs for dynamic geometry.
Intended learning outcomes - transferable/key skills and other attributes
• Critical thinking: Careful verification and proof of mathematical claims.
• Problem solving: Perceiving transformations as an option to convert the mathematical situation into a different situation, which is more convenient to deal with.
• Technical skills: Managing work with a computer program for dynamic geometry.
• Creating new ideas: A problem approach, whereby computer experimentation creates hypotheses and later either prove or refute them.
• Organization: Creating transparency in demanding geometric constructions using a computer.
Readings
• H. S. M. Coxeter, S. L. Greitzer: Geometry Revisited. Washington: MAA, 1967.
• G. Leversha: The Geometry of the Triangle, UK Mathematics trust, 2013
• C. Kimberling, Geometry in Action, a discovery approach using the Geometer's sketchpad, Key College Publishing, Emeryville, 2003.
• D. Palman: Trokut i kružnica. Zagreb: Element, 1994.
• D. Palman: Geometrijske konstrukcije. Zagreb: Element, 1996.
• D. Palman: Stereometrija. Zagreb: Element, 2002.
Prerequisits
There are none.
Additional information on implementation and assessment Mid-term testing:
Accomplished geometric constructions at the exercises in computer room. 10 %
Exams:
Written exam – problems 45 %
Oral exam 45 %
Each of the mentioned assessments must be assessed with a passing grade.
Passing grades of mid-term testing is required for taking the written exam – problems. Passing grade of written exam – problems is required to take the oral exam.
Written exam – problems can be replaced with at least two mid-term tests.
