Geometry in the Plane, in Space, and in the Complex Plane

This unit aims to explore plane geometry, its objects, and proofs. The unit also aims to introduce complex numbers. The sections on geometry and complex numbers each make up half of the unit.

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

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

- work on mathematical proofs

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

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

Geometry of the Plane

• Basic properties of lines, vectors, angles, and distances. Definitions of circles, triangles, transformations…
• Thales and Pythagoras. The Midpoint Theorem, the sum of the angles in a triangle.
• The three cases of triangle congruence: similar triangles. Characterization of parallelograms.
• Sine, cosine, and trigonometry. Generalized Pythagorean theorem and the law of sines in a triangle. Trigonometric formulas.
• Traditional competitions.
• Circle, positions of a 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; calculating the inverse; calculating square roots.
• Euler's formulas; imaginary exponential; argument and exponential notation;
• Trigonometry with Complex Numbers, Trigonometric Circle, Trigonometry Formulas.
• Calculation of the product and the inverse (in exponential notation); nth roots of unity for any complex number; sum of the nth roots of unity; solving quadratic equations.
• Isometries of the plane. Classification; complex forms of isometries of the plane. Homotheties. The use of complex numbers in geometry.

High school mathematics curriculum (particularly geometry), and at a minimum, the mathematics specialization in 11th grade and the mathematics specialization in 12th grade, or the supplementary mathematics elective.

Recommended prerequisites:

High school mathematics curriculum (particularly geometry), ideally with a mathematics specialization, or even an advanced mathematics track.

Hourly volumes:

            CM: 7:30 p.m.

            TD: 7:30 p.m.

            TP: 0

            Land: 0