Geometri Uncategorized

Euclidean Definitions

Pic of blossoming Rhododendron catawbiense
Rhododendron catawbiense

A line has length between two points but no width.

A point has no components i.e. it can’t be divided into parts.

A straight line traces the closest distance between two points.

A curved line has no straight segments.

A surface or a plane is  a completeky flat surface between its border lines. Lines constitutes the borders of planes.

Two straight lines are said to intersect each other in a point where they meet. The inclination between those lines is called the angle between the lines.

The point of intersection is called the vertex of the angle.

A circle is a planar figure limited by a curved line called the perimeter or periphery.

The distance from the center to the perimeter is called the radius and is everywhere on the circle  the same.

The diameter is a straight line from periphery to peiriphery through the center of the circle. It divides the circle into two equal parts-half-circles.

A triangle is a figure made up of three straight lines. These are called its  sides.

A quadrilateral figure is made up of four straight lines.

A polygonal figure is borders by more than four straight lines.

A triangle composed of three equal sides is called equilateral.

A triangle having two equal sides is termed isosceles triangle.

A right triangle has a perpendicular angle.


More Elements

Paeony ‘Karl Rosenfield’

There are two types of proof:

Direct proofs: Where it is proved (deduced or induced) from the basic axioms, definiyions or earlier proved theorems that the statement is correct.

Indirect proofs: The principle for this type of proofs is that it is proved that the opposite of the stated proposition cannot be true since this would lead to a contradiction or to something impossible.

In order to solve a geometrical problem it is often necessary to perform geometrical construction. This means e.g that the solution may be facilitated by

connecting given dots

extrapolate given lines

Bisect given lines or angles in halves,

Construct a line parallell to a given line

Constuct a line perpendicular to a given line.

construct circles etc.

Euclid then proceeds to define the equal-sign (2+3 =5), inequality sign ( A > B) and what is meant by addition (the sum of parts) and multiplication )repeated adition of the same number (A +A +A = 3A) and the difference between two quantitites A – B which eguals the quantity you must add to B to get A. .


Venus transit

Last Wednesday the planet Venus passed between the Sun and the earth. A rare astronomical event that will not happen again until 2117.

Venus transit