There are quite a lot of things. I suggest you buy a set of three textbooks for senior high school math league. The topic is very good. Try to spell it.
1, plane geometry
Basic requirements: master all the contents determined by the outline of junior high school mathematics competition. Supplementary requirements: area and area method. Several important theorems: Menelius Theorem, Seva Theorem, Ptolemy Theorem and siemsen Theorem. Several important extreme values: the point with the smallest sum of the distances to the three vertices of a triangle-fermat point. The center of gravity is the point where the sum of squares of the distances to the three vertices of a triangle is the smallest. The center of gravity is the point in the triangle where the distance product of three sides is the largest. Geometric inequality. Simple isoperimetric problem. Understand the following theorem: In a group of N-polygons with a certain perimeter, the area of a regular N-polygon is the largest. In a set of simple closed curves with a certain perimeter, the area of the circle is the largest. In a group of N-sided polygons with a certain area, the perimeter of the regular N-sided polygon is the smallest. In a set of simple closed curves with a certain area, the circumference of a circle is the smallest. Motion in geometry: reflection, translation and rotation. Complex number method and vector method. Planar convex set, convex hull and their applications.
2. Algebra
On the basis of the first test outline, other contents are needed: periodic functions and images of periodic functions with absolute values. Triple angle formula, some simple identities of triangle, triangle inequality. The second mathematical induction. Recursion, first and second order recursion, characteristic equation method. Function iteration, finding n iterations, simple function equation. N-element mean inequality, Cauchy inequality, rank inequality and their applications. Exponential form of complex number, Euler formula, Dimo theorem, unit root, application of unit root. Cyclic permutation, repeated permutation and combination, simple combinatorial identity. The number of roots of an unary n-degree equation (polynomial), the relationship between roots and coefficients, and the pairing theorem of imaginary roots of real coefficient equations. Simple elementary number theory problems should include infinite descent method, congruence, Euclid division, nonnegative minimum complete residue class, Gaussian function, Fermat's last theorem, Euler function, Sun Tzu's theorem, lattice points and their properties.
3. Solid geometry
Polyhedral angle, properties of polyhedral angle. Basic properties of trihedral angle and straight trihedral angle. Regular polyhedron, euler theorem. Proof method of volume. Sections, sections, and surface flat patterns will be made.
4. Plane analytic geometry
Normal formula of straight line, polar coordinate equation of straight line, straight line bundle and its application. The region represented by binary linear inequality. The area formula of triangle. Tangents and normals of conic curves. Power and root axis of a circle.
5. Others
Dove cage principle Exclusion principle. Extreme principle. Division of sets. Cover. Menelaus Theorem Ptolemy Theorem Existence and Properties of siemsen Line (siemsen Theorem). Seva theorem and its inverse theorem.