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COMMENT
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The first pedal of a triangle has as its vertices the feet of the perpendiculars of the original triangle. The (n+1)st pedal is the pedal of the n-th pedal.
From Fortschritte JFM 34.0551.02 on the Valyi paper: The triangle with corners the altitude bases of a given triangle ABC are
called pedal triangles. The pedal triangle of this triangle is the second
pedal triangle. Generally, we understand the n-th pedal triangle of the
triangle ABC to be the pedal triangle of the (n-1)th pedal triangle. The author
searches for and counts all triangles that are similar to their n-th pedal
triangle, where all mutually similar triangle are counted as one.
The number of these is psi(n)=2^n(2^n-1). The number of triangles for
which the n-th pedal triangle is the first that is similar to it
is chi(n) [...] where p_1,p_2,...p_n are the different prime factors
of n. The author ends with a table of those triangles that
are similar to their first, 2nd and 3rd pedal triangles.
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REFERENCES
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Alexander, J. C. The symbolic dynamics of the sequence of pedal triangles. Math. Mag. 66 (1993), no. 3, 147-158.
Ding, Jiu; Hitt, L. Richard; Zhang, Xin-Min, Markov chains and dynamic geometry of polygons. Linear Algebra Appl. 367 (2003), 255-270.
Hayashi, T. On the pedal triangles similar to the original triangles. Nieuw Archief (2) 10 (1912), 5-9. [Shows that there are 11 points whose pedal triangles are similar to the original triangle; those 11 points lie on a circle.]
Kingston, John G.; Synge, John L., The sequence of pedal triangles. Amer. Math. Monthly 95 (1988), no. 7, 609-620.
Ungar, Peter Mixing property of the pedal mapping. Amer. Math. Monthly 97 (1990), no. 10, 898-900.
Valyi, J., Ueber die Fusspunktdreiecke, Monatsh. f. Math. 14 (1903), 243-252.
de Vries, Jan, Ueber rechtwinklige Fusspunktdreiecke. Nieuw Archief (2) 9 (1910), 130-132. [The locus of those points that have rectangular pedal triangles with respect to a given triangle is determined by the three circles that cut the circumscribing circle orthogonally at two vertices of the triangle.]
Veldkamp, G. R. Classical geometry [Dutch], in Geometry, From Art to Science [Dutch], 1-15, CWI Syllabi, 33, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1993.
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