Important Formulae & Rules- Part - 1

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Important Formulae & Rules- Part - 1

Author: Leonardodvin

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Arithmetic-Geometric Means Inequality

If n is a positive integer and $a_1,a_2,\cdots , a_n$ are non-negative real numbers, then

$\frac {1}{n}\overset {n}{\underset {i=1}{\sum}}a_i\underline > (a_1a_2\cdots a_n)^{1/n},$

with equality if and only if $a_1=a_2=\cdots =a_n$ . This inequality is a special case of the power mean inequality.

Arithmetic-Harmonic Means Inequality

If $a_1,a_2,\cdots , a_n$ are positive numbers, then

$\frac {1}{n}\overset {n}{\underset {i=1}{\sum}}a_i\underline > \frac {1}{\frac {1}{n}\sum^n_{i=1}\frac {1}{a_i}},$

with equality if and only if $a_1=a_2=\cdots =a_n$ . This inequality is a special case of the power mean inequality.

Binomial Coefficient

$\dbinom{n}{k}=\frac {n!}{k!(n-k)!},$

the coefficient of x^k in the expansion of (x+1)^n

Cauchy-Schwarz Inequality

For any real numbers $a_1,a_2,\cdots , a_n$ and $b_1,b_2,\cdots , b_n$

$(a_1^2+a_2^2+\cdots +a_n^2)(b_1^2+b_2^2+\cdots +b_n^2)\underline >(a_1b_1+a_2b_2+\cdots +a_nb_n)^2,$

with equality if and only if a_i and b_i are proportional, $i=1,2,\cdots n.$

Ceva's Theorem and Its Trigonometric Form

Let AD, BE, CF be three cevians of triangle ABC. The following are equivalent:

AD,BE,CF are concurrent;
$\frac {|AF|}{|FB|}.\frac {|BD|}{|DC|}.\frac {|CE|}{|EA|}=1;$
$\frac {\sin \angle ABE}{\sin \angle EBC}.\frac {\sin \angle BCF}{\sin \angle FCA}.\frac {\sin \angle CAD}{\sin \angle DAB}=1$

Cevian
A cevian of a triangle is any segment joining a vertex to a point on the opposite side.

Chebyshev's Inequality

Let $x_1, x_2,\cdots, x_n$ and $y_1, y_2,\cdots, y_n$ be two sequences of real numbers such that $x_1\underline <x_2\underline <\cdots \underline <x_n$ and $y_1\underline <y_2\underline <\cdots \underline <y_n.$ Then

$\frac {1}{n}(x_1+x_2+\cdots x_n)(y_1+y_2+\cdots y_n)\underline < x_1y_1+x_2y_2+\cdots +x_ny_n.$
Let $x_1, x_2,\cdots, x_n$ and $y_1, y_2,\cdots, y_n$ be two sequences of real numbers such that $x_1\underline >x_2\underline >\cdots \underline >x_n$ and $y_1\underline >y_2\underline >\cdots \underline >y_n.$ Then

$\frac {1}{n}(x_1+x_2+\cdots x_n)(y_1+y_2+\cdots y_n)\underline > x_1y_1+x_2y_2+\cdots +x_ny_n.$

Chebyshev Polynomials

Let $\{T_n(x)\}^{\infty}_{n=0}$ be the sequence of polynomials such that T_0(x)=1,T_1(x)=x, and $T_{i+1}=2xT_i(x)-T_{i-1}(x)$ for all positive integers i. The polynomial T_n(x) is called the nth Chebyshev polynomial.

Circumcenter

The center of the circumscribed circle or sphere.

Circumcircle

A circumscribed circle.

Convexity

A function f(x) is concave up (down) on $[a,b]\subseteq R$ if f(x) lies under (over) the line connecting (a_1, f(a_1)) and (b_1,f(b_1)) for all

$a\underline < a_1 < x <b_1 \underline < b$

Concave up and down functions are also called convex and concave, respectively. If f is concave up on an interval [a, b] and $\lambda_1, \lambda_2, \cdots, \lambda_n$ are nonnegative numbers with sum equal to 1, then

$\lambda_1f(x_1)+\lambda_2f(x_2)+\cdots +\lambda_nf(x_n)\underline > f(\lambda_1x_1+\lambda_2x_2+\cdots +\lambda_nx_n)$

for any $x_1,x_2\cdots ,x_n$ in the interval [a,b] . If the function is concave down, the inequality is reversed. This is Jensen's inequality.

Cyclic Sum

Let n be a positive integer. Given a function f of n variables, define the cyclic sum of variables $(x_1, x_2,\cdots , x_n)$ as

$\overset {}{\underset {cyc}{\sum}}f(x_1,x_2,\cdots ,x_n)=f(x_1,x_2,\cdots ,x_n)+f(x_2,x_3,\cdots ,x_n,x_1)+\cdots +f(x_n,x_1,x_2,\cdots ,x_{n-1}).$

De Moivre's Formula

For any angle Î± and for any integer n,

$(\cos \alpha+i\sin \alpha)^n=\cos n\alpha+i \sin n\alpha$

From this formula, we can easily derive the expansion formulas of $\sin n\alpha$ and $\cos n\alpha$ in terms of $\sin \alpha$ and $\cos \alpha .$

Euler's Formula (in Plane Geometry)

Let O and I be the circumcenter and incenter, respectively, of a triangle with circumradius R and inradius r. Then

|OI|^2=R^2-2Rr

Excircles, or Escribed Circles

Given a triangle ABC, there are four circles tangent to the lines AB, BC, CA. One is the inscribed circle, which lies in the interior of the triangle. One lies on the opposite side of line BC from A, and is called the excircle (escribed circle) opposite A, and similarly for the other two sides. The excenter opposite A is the center of the excircle opposite A; it lies on the internal angle bisector of A and the external angle bisectors of B and C.

Extended Law of Sines
In a triangle ABC with circumradius equal to R,

$\frac {|BC|}{\sin A}=\frac {|CA|}{\sin B}=\frac {|AB|}{\sin C}=2R$

Gauss's Lemma
Let

$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_ax+a_0$

be a polynomial with integer coefficients. All the rational roots (if there are any) of p(x) can be written in the reduced form $\frac {m}{n}$ , where m and n are divisors of a_0 and a_n , respectively.

Gergonne Point

If the incircle of triangle ABC touches sides AB, BC and CA at F, D and E then lines AD, BE and CF are concurrent, and the point of concurrency is called the Gergonne point of the triangle.

Heron's Formula

The area of a triangle ABC with sides a, b, c is equal to

$[ABC]=\sqrt {s(s-a)(s-b)(s-c)},$

where

is the semi-perimeter of the triangle.

Homothety

Ahomothety (central similarity) is a transformation that fixes one point O (its center) and maps each point P to a point P' for which O, P, P' are collinear and the ratio |OP| : |OP'| = k is constant (k can be either positive or negative); k is called the magnitude of the homothety.

Homothetic Triangles

Two triangles ABC and DEF are homothetic if they have parallel sides. Suppose that AB||DE, BC||EF, and CA||FD. Then lines AD, BE and CF concur at a point X, as given by a special case of Desargues's theorem. Furthermore, some homothety centered at X maps triangle ABC onto triangle DEF.

Incenter
The center of an inscribed circle.

Incircle
An inscribed circle.

Kite

A quadrilateral with its sides forming two pairs of congruent adjacent sides. A kite is symmetric with one of its diagonals. (If it is symmetric with both diagonals, it becomes a rhombus.) The two diagonals of a kite are perpendicular to each other. For example, if ABCD is a quadrilateral with |AB|=|AD| and |CB|=|CD|, then ABCD is a kite, and it is symmetric with respect to the diagonal AC.

Lagrange's Interpolation Formula

Let $x_0, x_1,\cdots , x_n$ be distinct real numbers, and let $y_0, y_1,\cdots , y_n$ be arbitrary real numbers. Then there exists a unique polynomial p(x) of degree at most n such that $P(x_i)=y_i,i=0,1,\cdots ,n.$ This polynomial is given by

$P(x)=\overset {n}{\underset {i=0}{\sum}}\frac {y_i(x-x_0)\cdots (x-x_i-1)(x-x_i+1)\cdots (x-x_n)}{(x_i-x_0)\cdots (x_i-x_i-1)(x_i-x_i+1)\cdots (x_i-x_n)}$