Harmonic division

Points in harmonic division

It is said that reality $a \,$ is mean harmonic of $b \,$ and $c \,$ if $\ frac2a= \ frac1b+ \ frac1c$ .

Four points $A, B, C, D \,$ of a line are known as in harmonic division if $\ overline {AC}$ is mean harmonic of $\ overline {AB}$ and $\ overline {AD}$ ; that is to say

relation of Descartes : $\ frac2 {\ overline {AC}} = \ frac1 {\ overline {AB}} + \ frac1 {\ overline {AD}}$ .

One can still write this relation in the form $\ frac1 {\ overline {AC}} - \ frac1 {\ overline {AB}} = \ frac1 {\ overline {AD}} - \ frac1 {\ overline {AC}}$ i.e. $\ frac {\ overline {CB}} {\ overline {AB}} = \ frac {\ overline {cd.}} {\ overline {AD}}$ that one prefers to put in the form $\ frac {\ overline {AB}} {\ overline {AD}} = \ frac {\ overline {CB}} {\ overline {CD}}$ .

The quantity $\ frac {\ overline {AB}} {\ overline {CB}} \, \ frac {\ overline {AD}} {\ overline {CD}}$ which in this case takes the value -1 names the birapport or anharmonic ratio of the four points and is an interesting invariant in projective geometry.

See also : Anharmonic ratio

It is also said that B, D divides the segment harmonically because the reports/ratios $\ frac {\ overline {CB}} {\ overline {AB}}$ (division interior) and $\ frac {\ overline {cd.}} {\ overline {AD}}$ (division outside) are equal.

In this form, the relation is clearly symmetrical.

So divided harmonically then divides harmonically.

It is easily proven that has, B, C, D are in harmonic division if and only if one of the following relations is checked

relation of Newton : $IA^2 = IB^2 = \ overrightarrow {IC}. \ overrightarrow {ID}$ where I is the medium of;

relation of Mac-Laurin : $\ overrightarrow {AC}. \ overrightarrow {AD} = \ overrightarrow {AB}. \ overrightarrow {AJ}$ where J is the medium of.

Geometrical construction

Being given three points on a line,

A, B

and

I

one can build

J

such as

divides

harmonically as follows:

That is to say $M$ a point not aligned with the precedents; the parallel to $(MY)$ resulting from $B$ cut $(SEMI)$ in a point $I'$ ;

That is to say $J'$ such as $BJ'=-BI'$ then $(MJ')$ cut $(AB)$ in $J$ which is the sought point.

Harmonic beam of right-hand sides

Let us consider a beam of four lines $\ mathcal D_1, \ mathcal D_2 \ mathcal D_3, \ mathcal D_4$ exits of a point $O \,$ . It is supposed that a line $\ mathcal D \,$ the cut in points $M_1, M_2, M_3, M_4 \,$ forming a harmonic division, then it in will be the same for any line $\ mathcal Of \,$ .

This result is still true if the lines are parallel (beam resulting from a point ad infinitum in projective geometry).

This property thus depends only on the relative position of the right-hand sides on beam.

The beam of the four lines is then described as harmonic.

Demonstrations: harmonic Beam

Examples

To make

Bisectrices

Line of Euler

Complete quadrilateral

Orthogonal circles, combined points

That is to say $\ mathcal C$ a circle and a cord; That is to say $M~$ and $M'~$ two points aligned with $(AB) \,$ then the circle $\ mathcal C'$ of diameter $MM' \,$ is orthogonal with $\ mathcal C$ if and only if divides harmonically

It is said whereas $M$ and $M'$ are combined compared to the circle .

This property can extend if the points do not define a cord.

Two points $M, M'$ known as will be combined compared to a circle if the circle of diameter $MM'$ is orthogonal with $\ mathcal C$ .

PROPERTY: Two points are combined compared to a circle of center $O$ if and only if $\ overrightarrow {OM} \ overrightarrow {OM'} =R^2$ .

Either $O'$ the medium of $MM'$ (and thus the center of the new circle), and $T$ a point of intersection enters the two circles. By using $\ overrightarrow {O' M} = \ overrightarrow {O' Me}$ and $OT^2=R^2$ one obtains

$\ overrightarrow {OM} \ overrightarrow {OM'} = (\ overrightarrow {OT} + \ overrightarrow {TO'} + \ overrightarrow {O' Me}) (\ overrightarrow {OT} + \ overrightarrow {TO'} + \ overrightarrow {O' M}) =
R^2+2 \, \ overrightarrow {OT} \ overrightarrow {O' T},$

from where the result.

See now the article on the reciprocal Polar

Polar of a point compared to two lines

Definition: being given two lines D and of and two points M and Me distinct not located on these lines, the line (ME) meets D respectively and of out of P and P' distincts.
On says that M and are combined to Me harmonic compared to D and of if Me, P, P' forms a harmonic division.

Definition: being given two lines D and of distinct and convergent in an item I of the plan closely connected and a point M not located on these lines, the whole of combined harmonic of the point M compared to D and of is a line passing by I.
On calls it polar M compared to D and of . Construction of polar the : being given two lines D and of , convergent in an item I, and a point M not located on these lines, to place two points P and Q, distinct and different of I, on D and to plot two straight lines (MP) and (MQ). These lines respectively cut of in P' and Q'. One obtains the Complete quadrilateral MPP' Q' IQ. Its diagonals Δ = (PQ') and Δ' = (P' Q) are cut in J. line (IJ) is the polar M compared to D and of.

Demonstration : if M₁ is combined M compared to P and P' and combined M₂ M compared to Q and Q', the polar one of M compared to D and of is the line (M₁M₂); items I, M₁ and M₂ are alignés.
In the same way the polar one of M compared to Δ and Δ' is the line (M₁M₂); the points J, M₁ and M₂ are aligned and polar M compared to D and of is line (IJ).

See too

Polar reciprocal

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