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Binet’s Formula for Fibonacci Numbers
Cassini’s Identity for Fibonacci Numbers
Monty Hall Paradox in Probability
Chain Rule for Events in Probability
Classifying the Platonic Solids
A Criterion for Planar Graphs
Euler’s Characteristic Formula for Planar Graphs
Handshaking Lemma in Graph Theory
Stirling Partition Number of a Set
Balls & Urns, Sticks & Stones, Stars & Bars
Counting Surjections in Combinatorics
Counting Derangements in Combinatorics
Euler’s Totient Function Formula
Principle of Inclusion-Exclusion (PIE)
The Multinomial Theorem
The Binomial Theorem
Vandermonde’s Identity
Hockey Stick Identity
Sum of Squares of a Row of Pascal’s Triangle
Sum of a Row of Pascal’s Triangle
Symmetry in Pascal’s Triangle
Pascal’s Triangle and Pascal’s Identity
Multinomial Coefficients
Binomial Coefficients and Combinations
$k$-to-$1$ Correspondences
Counting Permutations
Counting the Power Set of a Finite Set
Friends and Strangers in Graph Theory
Strong Pigeonhole Principle and Reverse
Casework and Complementary Counting
What is Combinatorics?
Trigonometric Vector Cross Product
Vector Cross Product Derivation
Trigonometric Dot Product
Euclidean Triangle Inequality
Regular Octahedron: Surface Area & Volume
Regular Tetrahedron: Surface Area & Volume
Length of a Space Diagonal
Six Area Formulas for Triangles
Extended Law of Sines
Diagonals of a Cyclic Quadrilateral
Stewart’s Theorem for Triangles
Ceva’s Theorem for Triangles
Pitot’s Theorem
Cyclic Quadrilaterals
Power of a Point Theorem
Inscribed Angle Theorem
Converse of Thales’s Circle Theorem
Thales’s Circle Theorem
The Law of Cosines
Point-Line Distance Formula
Chebyshev’s Inequality
Schur’s Inequality
Cauchy-Schwarz Inequality
Arithmetic-Geometric Mean Inequality
Sophie Germain Identity
Difference and Sum of Powers
Vieta’s Polynomial Formulas
Polynomial Remainder and Factor Theorems
Complex Conjugates of Polynomial Roots
Polynomial Integer Root Theorem
Polynomial Rational Root Theorem
Why is $\deg(0)=-\infty$?
Square Root of a Complex Number
The Quadratic Formula
$n^{\text{th}}$ Roots of a Complex Number
De Moivre’s Formula
Complex Triangle Inequality
Triple Angle Trig Identities
Double & Half Angle Trig Identities
Pythagorean Theorem
Angle Bisector Theorem
Exterior Angle Theorem
Interior Angle Theorem
Sum of Powers of Positive Integers
Telescoping & Partial Fraction Decomposition
Geometric Series
Arithmetic Series Using Gauss’s Formula
Floor and Ceiling Sum Bounds
Hermite’s Floor Identity
Reverse Triangle Inequality
Real Triangle Inequality
Compound vs. Simple Interest
Bernoulli’s Inequality
Real Archimedean Property
Discrete Fubini’s Principle
Harmonic Series Divergence
Counting the Pages of a Book
Logarithmic Change of Base Identity
Rationalizing a Denominator
Unique Even and Odd Parts of a Function
Inverses in a Binary Operation are Unique
Identity Element of a Binary Operation is Unique
Cantor’s Paradox in Set Theory
Cantor’s Theorem in Set Theory
Bijective Function if and only if Inverse Exists
Bertrand Russell’s Paradox in Set Theory
Subset Relation is a Partial Order
Bounded Quantifiers in Logic
De Morgan’s Laws in Logic
The Principle of Explosion in Logic
Why Modus Ponens and Modus Tollens are Valid