Forward-Backward Proof Techniques.

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{2}A_{21} Base Sequence = 12735 String Sequence = 12735 - 2 - 21 **

Expressions Of Pj Problems

Forward-Backward Proof Techniques

Math

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

The right triangle XYZ of figure 121.1 has sides of lengths x and y, and hypotenuse of length z. Its area is z^{2}/4. Using the Forward-Backward proof techniques, proof that triangle XYZ is isosceles.

**The strings**:
S_{7}P_{2}A_{21} (Identity - Physical Property).
**The math**:

Pj Problem of Interest is of type *identity* (physical property). Proofs establish truths. So they are *identity* problems.

The Forward-Backward proof techniques are generally used to establish the *If A then B truth*. Where A and B are statements. The statements in the given problem are:

A: Right triangle XYZ with sides of length x and y, and hypotenuse of length z has an area of x^{2}/4.

B: Right triangle XYZ is isosceles.

Proof of Interest: If A then B (A implies B).

Forward Process Method:

Proof begins with assuming A is true then progresses stepwise with intermediary truths given the truth of A until B is established.

Truth of A:

Area =xy/2 = z^{2}/4 ------(1).

x^{2} + y^{2} = z^{2} (Pythagoras Theorem)-------(2)

So, xy/2 = (x^{2} + y^{2})/4

So, 0 = (x -y)^{2}

So, (x - y) = 0

So, x = y

So, B is established.

So If A then B (A implies B).

Backward Process Method:

Proof begins with assuming B is true then progresses stepwise with intermediary truths given the truth of B until A is established.

x = y.

So, (x -y)^{2} = 0.

So, x^{2} -2xy + y^{2} = 0.

So, z^{2} = 2xy

So, z^{2}/4 = xy/2 = area of XYZ

So, A is established.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Single Variable Functions

Conics

Ordinary Differential Equations (ODEs)

Vector Spaces

Real Numbers

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings