# The Mathematics

## Angles Opposite to Equal Sides of an Isosceles Triangle are Equal

Here we will prove that in an isosceles triangle, the anglesopposite to the equal sides are equal. Solution: Given: In the isosceles ∆XYZ, XY = XZ. To prove ∠XYZ = ∠XZY. Construction: Draw a line XM such that it bisects ∠YXZ and meets the side YZ at M. Proof:           Statement 1. …

## Application of Congruency of Triangles

Here we will prove some Applicationof congruency of triangles. 1. PQRS is a rectangle and POQ an equilateral triangle. Provethat SRO is an isosceles triangle. Solution: Given: PQRS is a rectangle. POQ is an equilateral triangle to prove ∆SOR is an isosceles triangle. Proof:           Statement           Reason 1. …

## Bisectors of the Angles of a Triangle Meet at a Point

Here we will prove that the bisectors of the angles of atriangle meet at a point. Solution: Given In ∆XYZ, XO and YO bisect ∠YXZ and ∠XYZrespectively. To prove: OZ bisects ∠XZY. Construction: Draw OA ⊥ YZ, OB ⊥ XZ and OC ⊥ XY. Proof:           Statement 1. In ∆XOC and …

## Point on the Bisector of an Angle

Here we will prove that any point on the bisector of anangle is equidistant from the arms of that angle. Solution: Given OZ bisects ∠XOY and PM ⊥ XO and PN ⊥ OY. To prove PM = PN. Proof:             Statement 1. In ∆OPM and ∆OPN, (i) ∠MOP = ∠NOP. …

## Affine Varieties – The Math

Definition: Let \$K\$ be a field and let \$X subseteq mathbb{A}^n(K)\$ be an affine algebraic set. Then \$X\$ is said to be Reducible if there exists affine algebraic sets \$X_1\$ and \$X_2\$ where \$X_1, X_2 neq emptyset\$ and \$X_1, X_2 neq X\$ and such that \$X = X_1 cup X_2\$. An affine algebraic set \$X\$ …

## The Coordinate Ring of an Affine Variety

Recall from the Affine Varieties page that an affine variety is simply an irreducible affine algebraic set. When we have a nonempty affine variety \$V\$ we can consider what is called the coordinate ring of \$V\$ which we define below. Definition: Let \$K\$ be a field and let \$V subset mathbb{A}^n(K)\$ be an nonempty affine …

## The Rational Function Field of an Affine Variety

Definition: Let \$K\$ be a field and let \$V subseteq mathbb{A}^n(K)\$ be a nonempty affine variety. The Rational Function Field of \$V\$ is defined as \$K(V) = left { frac{f}{g} : f, g in Gamma(V), g neq 0 right }\$. Elements of \$K(V)\$ are called Rational Functions. The rational function field of \$V\$ is also …

## Local Rings – The Math

Definition: A Local Ring is a ring \$R\$ that has a unique maximal ideal. Recall that a proper ideal \$I\$ of a ring \$R\$ is a maximal ideal if there exists no other proper ideals of \$R\$ containing \$I\$. In general, maximal ideals need not be unique. For example, consider the ring of integers \$mathbb{Z}\$. …

## Discrete Valuation Rings – The Math

Definition: A Discrete Valuation Ring (DVR) is an integral domain \$R\$ with the following properties:1) \$R\$ is a Noetherian ring.2) \$R\$ is a local ring.3) The unique maximal ideal of \$R\$ is a principal ideal. Recall that an ideal \$I\$ of a ring \$R\$ is a principal ideal if it is generated by a single …

## Polynomial Forms – The Math

Definition: Let \$R\$ be a ring. A polynomial \$F in R[x_1, x_2, …, x_n]\$ is said to be a Form of Degree \$d\$ if every term in \$F\$ is of degree \$d\$. For example, the following polynomials are forms of degree \$2\$, \$3\$, and \$4\$ respectively: (1) begin{align} quad F(x, y, z) = x^2 + …