14700[EMHL] Re: Locus of Simson line intersection point
- Jan 6, 2007Dear 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
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
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