- Dear Jean-Pierre,

Thank you very much.

> > many thanks for your nice mails about this

*********

> problem.

> > You've probably noticed that when the angle of the

> Simson lines is

> > Pi/3, we get a nice trifolium inscribed in the

> circle(N,R) N = NP-

> center

> > With N as pole and a tangent at a cusp of the

> Steiner deltoid as

> polar

> > axis, the trifolium has polar equation rho =

> R.cos(3.theta)

>

> Some remarks about this trifolium :

> The vertices - contact points with the circle(N,R) -

> are the vertices

> of the equilateral triangle bounded by the Simson

> lines of the

> vertices of the circumnormal triangle; these Simson

> lines are the

> common tangents of the Steiner deltoid and the

> NP-circle at the

> points where they touch each other.

These points are found by drawing from N parallells

to the bisectors of Morley's triangle.

> These vertices

********

> are homothetic of

> the cusps of the Steiner deltoid in (N,2/3)

> The rectangular circumhyperbola through N intersects

> the circle (N,R)

> at 4 points; one of them is the antipode of N on the

> hyperbola; the

> three other ones are the vertices above.

Very interesting.

Happy New Year to you and all Hyacinthists.

Best regards

Nikos Dergiades.

___________________________________________________________

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μηνυμάτων http://login.yahoo.com/config/mail?.intl=gr - Dear Quang Tuan Bui

in order to generalize the Steiner deltoid, we can look at the locus of

the orthopoles of the lines tangent to a given circle with center O; we

get the cycloidal curves we were talking about.

An other way to generalize is the following :

For M lying on the circumcircle, let A1, B1, C1 lie on the sidelines of

ABC such as the three oriented line angles (BC, MA'), (CA, MB'), (AB,

MC') have the same fixed value w.

Then A1,B1,C1 lie on a same line and this line envelops a deltoid

touching the sidelines of ABC. This deltoid is "centered" at U, lying

on the perpendicular bisector of OH and such as <NUO =w.

The ratio of a similitude mapping the Steiner deltoid to this deltoid

is 1/sin(w)

Some properties of this deltoid and the connection with Mac Beath's

works have been discussed in Hyacinthos.

See, for instance #9934, #9947+ and the file Mac Beath.pdf

Friendly. Jean-Pierre