Geometry in the plane, space and complex plane

This UE aims at working on the geometry of the plane, its objects but also the demonstrations. The UE also aims at introducing complex numbers. The parts geometry and complex numbers represent each half of the UE.

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

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

- work on mathematical demonstration

- introduction of complex numbers, geometric interpretation, calculation with complex numbers

The course is based on the notions seen in high school. It is not an axiomatic approach. The parts geometry and complex numbers represent each half of the UE.

Geometry of the plane

• Elementary properties of lines, vectors, angles, distance. Definitions of circles, triangles, transformations...
• Thales and Pythagoras. Midpoint theorem, sum of angles in a triangle.
• The three cases of equality of triangles, similar triangles. Characterization of parallelograms.
• Sine, cosine and trigonometry. Generalized Pythagorean theorem and sine theorem in a triangle. Form of trigonometry.
• Classic competitions.
• Circles, positions of a line relative to a circle, tangents. Inscribed and circumscribed circle. Inscribed angle theorem.

Complex numbers

• Complex numbers: algebraic notation; geometric viewpoint, affixes, operations ;
• Conjugate and modulus; calculating the inverse; calculating square roots.
• Euler formulas; imaginary exponential; argument and exponential notation ;
• Trigonometry with complexes, Trigonometric circle, Trigonometry form.
• Calculation of product and inverse (in exponential notation); n-th roots of unity, of any complex; sum of n-th roots of unity; solution of second degree equations.
• Isometries of the plane. Classification, complex form of the isometries of the plane. Homotheties. Use of complex numbers in geometry.

High school mathematics program (especially geometry), and at least a first year mathematics speciality and a final year mathematics speciality or additional mathematics option.

Recommended prerequisites:

High school mathematics program (especially geometry), ideally specializing in mathematics, or even an expert mathematics option.

Hourly volumes:

            CM : 19,5 h

            TD : 19,5 h

            TP : 0

            Land : 0