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