Kategorier

## Heron’s formula

Heron who invented rhe first steam device also came up with a geometry formula.

The circumference p of a triangle is related to the sides of of the triangle a,b and c and the trianglearea T through

T = √ (p(p-a)(p-b)(p-c)

Kategorier

## Some geometric theorems: Menelao’s, Ceva’s, Simson’s line and Stewart’s

The Italian mathematician Giovanni Ceva’s published in 1678 Menalao’s theorem and that which is credited to him.

Use Menlao’s theorem for collinearity and Ceva’s theorem för concurrency.

Menelao’s theorem (named after MEnelaos from Alexandria  100 A.D) states that if points P, Q and R are taken on sides AC, AB or BC of triangle ΔABC  these points are collinear if and only if

AQ/QB · BR/RC · CP/PA = -1.

Ceva’s theorem says that three lines drawn from the vertices  A, B and C of ABC meetinng the opposite sides in points L, M, N respectively , are congruemt if and only if

AN/NB · BL/RC ·  CP/PA =1. Ceva’s theoremSimson’s line: The feet of the perpendiculars drawn from any point on the circumference of a circumscribed circle to the sides of the triangle are collinear.

Kategorier

## Kurrebo, Urshult, An Eden in Sweden The public garden and apple orchard in Urshult Sweden.
The northernmost applefarm in the world.With a fantastic rosarium. The seed company Nelson also exhibits its assortiment of bulbs and seeds here. Fantastic fragrance!

Kategorier

## Definition of groups

A group in mathematics has the following properties:

There exist a set of elements p, q, r, …  and a binary operation which applied to p, q gives pq.

The set is closed under this single-valued operation.

1.  The associative law: p(qr) = (pq)r.
2. Identity law: p i = i p.
3. Inverse law:   there exist p’ such that p p’ = p’ p

If the group axioms are full-filled i is unique and the inverse is also unique.

Kategorier

## Normalized coordinates

If A, b, c are three non-linear points, any vector P inside the triangle ABC may be expressed in terms of vectors A, B C thus:

P =xA + yB + zC   with x + y + z = 0

and x,y,z being uniquely determined.

This is called barycentric or normalized

coordinates.

Kategorier

## Collinearity

Theorem:
If A, B and C are collinear points, then the real numbers x, y, z not all zero can be found such that
x + y + z = 0 xA + yB + zC = 0.
and also the inverse of this thorem is true: If three such numbers not all zero can be ound then the points are collinear.

Conversely
Theorem:
If A, B, C are three given points which are not collinear and we can find three real numbers x, y, z such that x+y+z = 0 and xA + yB + zC = 0,
then we must have x=y=z=0.

Kategorier

## More vector algebra

A clever way to express the equation of a line is to use the parameter form.

If C is any point on the line determined by two points a and B then we may always write

C = (1-t) A + t B

where the ratio of the real numbers t/(t-1) = AC / CB. t is the parameter with values from 0 to 1.

I give a proof for this statement:

Let AC = t AB.

This equation can be translated into

C-A = t (B-A)  → C = (1 – t) A + t B.

It is also a fact that

AB = AC + CB

AC = t (AB) → AC = t(AC + CB)  → (1 -t)AC and t/(t-1) = AC/CB

Q.E.D.

Kategorier

## Venus transit in H- alpha

Picture of Venus transit photographed with an Hα- filter.

λ = 6566 A. Resolution 1 A.
In this picture it is possible to cpmåare the size of the Earth to that of the Sun since Venus is of the same size as the Earth.

Kategorier

## Linear dependence The vectors P1, P2, P3, … are said to be linearly dependent if the real numbers k1, k2, k3, .. not all zero can be found so that

k1P1 + k2P2 + ….. + knPn = 0.

Since it is possible to solve for e.g. P1= -k2P2/k1-k3P3/k1-….

This means that all the vectors lie on the same line through the origin.

Conversely, if two vectors lie along the same line they are linearly dependent vectors.

A set of vectors which are not dependent are said to be

linearly independent.

An example of linearly independent vectors are E1 = (1,0) and E2 = (0,1).

They form the basis for the two dimensional vector-space. In favt any two linearly-independent vectors can form the basis of a two dimensional vector space. The dimension is identical to the number of vectors necessary.

Two geometric formulae:

Menelaos theorem: A line cuts the sides BC, CA and AB of a triangel in the points L, M and N respectively. If L = xB + x’C + y’A, M = yC + y’A and

N = zA + z’B, where x +x’ = y + y’ = z +z’ = 1,

then xyz = -x’y’z’.