Geometry in the plane, space, and the complex plane

This unit aims to cover plane geometry, its objects, and proofs. The unit also aims to introduce complex numbers. The geometry and complex numbers sections each represent half of the unit.

- objects of plane geometry: points, lines, vectors, angles, circles, triangles, etc.

- geometric transformations of the plane: symmetries, homotheties, rotations, translations.

- work on mathematical proof

- introduction to complex numbers, geometric interpretation, calculations with complex numbers

The course builds on concepts covered in middle school/high school. It is by no means an axiomatic approach. The geometry and complex numbers sections each represent half of the course.

Plane geometry

• Basic properties of lines, vectors, angles, and distances. Definitions of circles, triangles, transformations, etc.
• Thales and Pythagoras. Midpoint theorem, sum of angles in a triangle.
• The three cases of triangle equality, similar triangles. Characterization of parallelograms.
• Sine, cosine, and trigonometry. Generalized Pythagorean theorem and sine theorem in a triangle. Trigonometry formula.
• Traditional competitions.
• Circle, positions of a straight line relative to a circle, tangents. Inscribed and circumscribed circles. Inscribed angle theorem.

Complex numbers

• Complex numbers: algebraic notation; geometric perspective, affix, operations;
• Conjugate and modulus; inverse calculation; square root calculation.
• Euler's formulas; imaginary exponential; argument and exponential notation;
• Trigonometry with complex numbers, trigonometric circle, trigonometry formula sheet.
• Calculation of the product and inverse (in exponential notation); nth roots of unity, of any complex number; sum of nth roots of unity; solving quadratic equations.
• Isometries of the plane. Classification, complex form of isometries of the plane. Homothety. Use of complex numbers in geometry.

High school mathematics program (particularly geometry), and at a minimum, a specialization in the junior year and a specialization in mathematics in the senior year, or an additional mathematics option.

Recommended prerequisites:

High school mathematics program (particularly geometry), ideally with a specialization in mathematics or even an advanced mathematics option.

Hourly volumes:

            CM: 19.5 hours

            Tutorial: 19.5 hours

            TP: 0

            Land: 0