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Topic: Harmonic set of points internal/external division NEED HELP
Replies: 4   Last Post: Nov 7, 2012 8:07 AM

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Dan Cass

Posts: 442
Registered: 12/6/04
Re: Harmonic set of points internal/external division NEED HELP
Posted: May 1, 2011 11:34 AM
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> In the James Smart text 'Modern Geometries' on page
> 214 fig 5.4 A Harmonic set of points is defined as
> follows:
> "In Euclidean Geometry, a harmonic set of points (or
> harmonic range) is defined to be a set of four
> collinear points such that the segment determined by
> two of them is divided internally and externally in
> the same ratio."
> To me this is not plain English. What does it mean
> to say: "the segment determined by two of them?" And
> mainly I don't understand this concept of dividing a
> segment externally.

suppose the points in increasing order are a,b,c,d.
Then b divides [a,c] internally in the ratio
[1] (b-a)/(c-b).
And d divides [a,c] externally in the ratio
[2] (d-a)/(d-b).
For certain choices of a,b,c,d. the ratios [1] and [2]
might be the same. In that case one has a harmonic set of points.

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