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**Law of Cosines**

In a triangle ABC,

and analogous equations hold for and .

**Median formula**

This is also called the **length of the
median formula.** Let AM be a median in triangle
ABC. Then

**Minimal Polynomial**

p<>. We call a polynomial p(x) with integer
coefficients **irreducible** if p(x) cannot be written
as a product of two polynomials with integer coefficients neither
of which is a constant. Suppose that the number is a root of a
polynomial q(x) with integer coefficients. Among all polynomials
with integer coefficients with leading coefficient 1 (i.e., monic
polynomials with integer coefficients) that have as a root, there
is one of smallest degree. This polynomial is the **minimal
polynomial** of Let p(x) denote
this polynomial. Then p(x) is irreducible, and for any other
polynomial q(x) with integer coefficients such that the polynomial p(x) divides q(x); that is, q(x) =
p(x)h(x) for some polynomial h(x) with integer
coefficients.

**Orthocenter of a
Triangle**

p<>. The point of intersection of the
altitudes.

**Periodic Function**

p<>. A function f (x) is periodic with
period T > 0 if T is the smallest positive real number for
which

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%{font-family:verdana}for all x.

**Pigeonhole Principle**

p<>. If n objects are distributed among k
< n boxes, some box contains at least two objects.

**Power Mean Inequality**

p<>. Let be any positive numbers for which
For positive numbers we define

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%{font-family:verdana}where t is a non-zero real number. Then

for

**Rearrangement
Inequality**

p<>. Let
be real numbers, and let be any permutations of
Then

with equality if and only if or

**Root Mean Squareâ€"Arithmetic Mean
Inequality**

p<>. For positive numbers

The inequality is a special case of the
**power mean inequality.**

**Schurâ€™s Inequality**

p<>. Let x, y, z be non-negative real
numbers. Then for any r > 0,

Equality holds if and only if x = y
= z or if two of x, y, z are equal and the third is equal to 0.

The proof of the inequality is rather simple. Because the
inequality is symmetric in the three variables, we may assume
without loss of generality that Then the given
inequality may be rewritten as

and every term on the left-hand side
is clearly nonnegative. The first term is positive if x > y, so
equality requires x = y, as well as which gives either or

**Sector**

p<>. The region enclosed by a circle and two
radii of the circle.

**Stewartâ€™s Theorem**

p<>. In a triangle ABC with cevian AD, write a = |BC|, b = |CA|, c =
|AB|, m = |BD|, n = |DC|, and d = |AD|. Then

This formula can be used to express
the lengths of the altitudes and angle bisectors of a triangle in
terms of its side lengths.

**Trigonometric
Identities**

p=.

**Viete's Theorem**

p<>. Let be the roots of polynomial

where and
Let be the sum of the products
of the
taken k at a time. Then

that is,

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