Geometry in the plane, space and complex plane

This UE aims to work on plane geometry and its objects, as well as demonstrations. It also introduces complex numbers. The geometry and complex numbers sections each account for half of the UE.

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

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

- work on mathematical demonstration

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

The course is based on notions learned in secondary school. It is by no means an axiomatic approach. Geometry and complex numbers each account for half of the course.

Plane geometry

• Basic properties of straight lines, vectors, angles and distances. 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. Trigonometry formulas.
• Classic competitions.
• Circles, positions of a line relative to a circle, tangents. Inscribed and circumscribed circles. Inscribed angle theorem.

Complex numbers

• Complex numbers: algebraic notation; geometric point of view, affixes, operations ;
• Conjugate and modulus; inverse calculation; square root calculation.
• 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.
• Plane isometries. Classification, complex form of plane isometries. Homotheties. Use of complex numbers in geometry.

High-school mathematics program (including geometry), and at least a première speciality and a terminale mathematics speciality or complementary mathematics option.

Recommended prerequisites :

High school mathematics program (especially geometry), ideally with a mathematics specialization or even an expert mathematics option.

Hourly volumes :

            CM: 19.5 h

            TD: 19.5 h

            TP: 0

            Land: 0