Geometry I

Syllabus

• Coordinates on the line, plane, and space; orientation (ordering, right-hand rule); distance and angle measures; directed angles.
• Relative position of geometric objects: parallelism, perpendicularity, angles; orthogonal projections.
• Isometries in plane and space: translations, rotations, reflections; similarity transformations and homotheties.
• Directed segments and vectors; vector operations and coordinates; bases and coordinate systems; basis vectors.
• Dot product, cross product, and scalar triple product; coordinate formulas and geometric meaning; classical vector identities.
• Equations of lines (2D/3D) and planes; link between coefficients and geometry; equations of circles and spheres.
• Centers of mass for weighted point systems; ratios and barycenters; grouping of weights.
• Convex sets and convex hull: definition, uniqueness; representations via convex combinations.
• Polylines and polygons; angle sums; convex polygons as convex hulls or intersections of half-planes.
• Polyhedra: Euler’s formula for convex polyhedra; dihedral/solid angles; regular polygons and regular polyhedra (classification).