We aim at self-contained proofs that are as short, clear and simple as possible. At times, they may be somewhat artificial. The links lead to pdf files that can be downloaded. 

• Solving the cubic and quartic equations (2 pages, added 2025-05-15)
• The theorems of Pappus and Pascal (6 pages, added 2025-05-16) Elementary straightforward proofs with determinants.
• Solving the linear differential equation of the second order with variable coefficients (5 pages, added 2025-05-22) Estimates for repeated integrals. Eliminating the first-order derivative. Two independent solutions given by explicit series in which each new term requires two more integrations. Ready-made formulas for reduction of order and variation of constants.
• Wallis and Stirling inequalities (5 pages, added 2025-05-23) Wallis's product and Stirling's formula are given in the quantitative form of double-sided inequalities.
• Euler's rotation theorem (5 pages, added 2025-05-28) Rotation matrices. Characterization of a rotation around an axis through the origin by its matrix. Euler's theorem on the composition of rotations. 
• Duplicating the sphere (13 pages, added 2025-06-08) Elaboration of Robinson's theorem (here): a solid sphere can be split in four pieces and a point on its surface in such a way that, if the pieces are rotated and the point translated, two copies of the original sphere are formed. Introduction. Four labels and their relations. Partitioning a spherical surface in equivalence classes. Partitioning a spherical surface in four pieces. Partitioning a spherical surface in four pieces and a point. Partitioning a solid sphere in four pieces and a point on its surface. Appendix: independent rotations. Two independent rotations. Four independent rotations.
• Hidden order (3 pages, added 2025-06-13) Erdös-Szekeres theorem on monotone subsequences in a finite sequence. Existence of monotone subsequences in any infinite sequence.
• de l'Hospital's rule (4 pages, added 2025-06-25) The 75 cases are reduced to a single property: bounds for f'/g' are also bounds for f/g (give or take an 𝜀).
• Fundamental Theorem of Algebra (2 pages, added 2025-06-29) Simple proof, using nothing beyond continuity.
• Symmetric polynomials (4 pages, added 2025-07-07) Fundamental theorem on symmetric polynomials: Every symmetric multivariate polynomial  P(x_1,x_2,...,x_n) can be expressed, in a unique way, as a polynomial Q(s_1,s_2,...,s_n) in the elementary symmetric functions of x_1,x_2,...,x_n. Elementary symmetric functions of the roots of univariate polynomials (very elementary stuff).
• Transcendence of 𝜋 (6 pages, added 2025-07-12) Transcendental numbers. Trancendence of 𝜋i and 𝜋. Appendix: complex mean-value theorem. By the latter, we avoid the integrals present in Ivan Niven's 1939 paper (here). The original theorem by Lindemann (1882) settled the question of squaring the circle.