--- In

Hyacinthos@yahoogroups.com, Nikolaos Dergiades

<ndergiades@y...> wrote:

> Hence we have the following construction of X(37).

> We extend the sides AB, AC to BC', CB' such that

> BC'=BC=CB'= a.

> From C' we draw C'B" ||= CB'= a and

> from B' we draw B'C" ||= BC'= a

> The segments BC", CB", B'C' have the same midpoint A*.

> Similarly define the points B*, C*.

> The distance of B from AC is asinC.

> The distance of C" from AC is asinA

> Hence the distance of A* from AC is a(sinA+sinC)/2.

> Similarly

> the distance of A* from AB is a(sinA+sinB)/2.

> This means that the line AA* passes through the point

> with trilinears (b+c : c+a : a+b) and this is X(37).

> Hence the triangles ABC, A*B*C* are perspective

> and the perspector is X(37).

Nikolaos's construction leads to many other perspectors.

Using the following slightly changed notation:

Let Ca be the point on AB such that BCa = BC

(with B between A and Ca)

Define Cb, Ab, Ac, Bc and Ba similarly

A*, B* and C* are the midpoints of CaBa, AbCb and BcAc

A1 is the intersection of AcBa and CaAb

Define B1 and C1 similarly

CaBa, AbCb and BcAc form a triangle A2B2C2

BcCb, AcCa and BaAb form a triangle A3B3C3

A4 is the intersection of CbAc and BcCa

Define B4 and C4 similarly

we have the following perspectivities

1) Triangle ABC and A*B*C* ==> Perspector X37

2) Triangle ABC and A1B1C1 ==> Perspector X8

3) Triangle ABC and A2B2C2 ==> Perspector X65

4) Triangle ABC and A3B3C3 ==> Perspector X65

5) Triangle A2B2C2 and A*B*C* ==> Perspector X210

6) Triangle A1B1C1 and A2B2C2 ==> Perspector not in ETC

7) Triangle A1B1C1 and A3B3C3 ==> Perspector not in ETC

8) Triangle ABC and A4B4C4 ==> Perspector X8

9) Triangle A2B2C2 and A4B4C4 ==> Perspector not in ETC

10) Triangle A3B3C3 and A4B4C4 ==> Perspector not in ETC

...

Best wishes for 2005

Eric

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