four are on the same line and the

(HA/HB) = - (GA/GB),

i. e. G, H divide internally and externaly the segment AB into the same ratio.

We often write:

(HA/HB):(GA/GB)=-1

and say that the

(A,B,H,G) = -1.

The basic example of harmonic conjugate points is shown in the figure below.

First equation results by applying Menelaus (see Menelaus.html ) to triangle ABC and its secant (GEF).

Second equation results by applying Ceva (see Ceva.html ) to the same triangle and point D.

Menelaus for triangle ABC and secant GEF: (GA/GB)(FC/FB)(EA/EC)=1,

Ceva for triangle ABC and point D (see Ceva.html ): (AH/HB)(BF/FC)(EC/EA)=-1.

Multiplying the two equations:

(GA/GB)(AH/HB)=-1, i.e. G, H are

This implies that for fixed H and D moving on line CH all the corresponding lines GEF pass

through the fixed point G,

Notice that G, K are also harmonically conjugate to E, F. In fact, the same arguments

apply to triangle CEF, Menelaus secant GAB and Ceva point D.

Thus also (E,F,G,K) = -1.

There is though also another reason, why G, K are harmonically conjugate to E, F and

this is that they are cut off on line GF by a

(CA, CB, CG, CH), see Harmonic_Bundle.html .

Then the relation:

HA/HB = - GA/GB => HA*GB + GA*HB = 0 =>

(a-h)(b-g)+(a-g)(b-h) = 0 => 2(ab+gh) = (a+b)(g+h).

2) Taking A as the origin of coordinates:

(a=0) => 2gh = b(g+h) i.e. 2/AB = (1/AG) + (1/AH),

i.e AB is the mean harmonic of AG and AH.

3) Taking the middle I/J of AB/GH as origin of coordinates (a+b=0, resp. g+h=0) we get at

Newton's relations:

IA² = IB² = IG*IH, JH² = JG² = JA*JB.

4) Relation (2) implies

2AH*AG = AB(AH+AG) = AB*(2AJ) => AH*AG = AB*AJ.

5) Relations (3) imply:

JA/JH = JH/JB = (JH-JA)/(JB-JH) = AH/BH, and JA/JG = JG/JB = (JA-JG)/(JG-JB) = GA/GB.

Hence

JA/JB = (JA/JH)*(JH/JB) = (GA/GB)².

i.e. if G and H divide harmonically AB in ratio k, then the middle J of GH divides AB in ratio

k². See Apollonian_Circles.html and ApollonianBundle.html .

Menelaus.html

Ceva.html

Harmonic_Bundle.html

Apollonian_Circles.html

ApollonianBundle.html

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