Antreas recently wrote that

>George Kapetis, the author of "Triangle Geometry" vol. I (in Greek), told me

>once (in telephon) that the second volume of his book will contain (among

>other things) "Lemoine Transformation".

I have remained curious about this transformation and about some

inconsistencies in Lemoine's use of it. And, also, about the link with

extraversion. To recap: Lemoine's "transformation continue en A" maps

a,b,c to a,-b,-c and s,sa,sb,sc to -sa,-s,sc,sb respectively, and so on

(whereas Conway's extraversion (in A) maps a,b,c to -a,b,c ...).

In the paper I have quoted recently, Lemoine transforms various points by

applying the above mapping to their trilinear coordinates. For example,

the inner Soddy centre {(a+ra/a)::} maps to {(a+r)/a:(b+rc)/b:(c+rb)/c}.

But in the case of the Nagel point {sa/a::} which should - according to the

above - map to {s/a:sc/b:sb/c}, Lemoine gives {-sa/a:sc/b:sb/c}. This is,

of course, the more preferable Nagel extraversion, but the inconsistency is

puzzling.

One explanation might be the reminder that Lemoine transformation (as,

indeed, extraversion) is extrinsic, ie dependent on the coordinate system.

Thus if a point with barycentric coordinates [x:y:z] maps to [x':y':z']

then the trilinear version {x/a:y/b:z/c} maps to {x'/a:-y'/b:-z'/c}. A

barycentric version of this is [-x':y:'z']. I note that when applied to

barycentrics the Lemoine transformation is equivalent to extraversion; but

that when applied to trilinears (as originally intended) it yields the

harmonic of the (barycentric) extraversion.

There is, I think, a further difficulty in the case of Lemoine's

transformation of the Soddy centres. These are defined as centres of

circles touching the three circles with centres A,B,C and radii sa,sb,sc.

Lemoine maps the latter into circles with centres A,B,C and radii -s,sc,sb

(and extraversion yields the same circles). He then assumes that the Soddy

centres map into the centres of these circles. Now drawing a figure (the

simplest case takes ABC equilateral) verifies that the x-coordinate of the

inner Soddy centre must be negative. But, according to Lemoine, the inner

Soddy centre maps to a point with (trilinear) cooordinates

{(a+r)/a:(b+rc)/b:(c+rb)/c} whose components are all positive. The

harmonic of this yields the appropriate negative coordinate: extraversion

rules OK?

Dick Tahta