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Law of Cosines
In a triangle ABC,
This is also called the length of the median formula. Let AM be a median in triangle ABC. Then
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.
p<>. A function f (x) is periodic with period T > 0 if T is the smallest positive real number for which
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
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
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
p<>. The region enclosed by a circle and two radii of the circle.
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.
p<>. Let be the roots of polynomial
where and Let be the sum of the products of the taken k at a time. Then
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