Hofstadter points
Let "ABC" be a given triangle. Let "r" be a positive real constant. Rotate the line segment "BC" about "B" through an angle "rB" towards "A" and let "LBC" be the line containing this line segment. Next rotate the line segment "BC" about "C" through an angle "rC" towards "A". Let "L'BC " be the line containing this line segment. Let the lines "LBC" and "L'BC " intersect at "A"("r"). In a similar way the points "B"("r") and "C"("r") are constructed. The triangle whose vertices are "A"("r"), "B"("r"), "C"("r") is the Hofstadter "r"-triangle (or, the "r"-Hofstadter triangle) of triangle "ABC". The trilinear coordinates of the vertices of the Hofstadter "r"-triangle are given below:
"A"("r") = ( 1 , sin "rB" / sin (1 − "r")"B" , sin "rC" / sin (1 − "r")"C" )
"B"("r") = ( sin "rA" / sin (1 − "r")"A" , 1 , sin "rC" / sin (1 − "r")"C" ) 
"C"("r") = ( sin "rA" / sin (1 − "r")"A" , sin (1 − "r")"B" / sin "rB" , 1 ) For a positive real constant "r" > 0, let "A"("r") "B"("r") "C"("r") be the Hofstadter "r"-triangle of triangle "ABC". Then the lines "AA"("r"), "BB"("r"), "CC"("r") are concurrent. The point of concurrence is the Hofstdter "r"-point of triangle "ABC". The trilinear coordinates of Hofstadter "r"-point are given below. ( sin "rA" / sin ( "A" − "rA") , sin "rB" / sin ( "B − "rB" ) , sin "rC" / sin ( "C" − "rC") ) The trilinear coordinates of these points cannot be obtained by plugging in the values 0 and 1 for "r" in the expressions for the trilinear coordinates for the Hofstdter "r"-point. Hofstadter zero-point is the limit of the Hofstadter "r"-point as "r" approaches zero. Hofstadter one-point is the limit of the Hofstadter "r"-point as "r" approaches one. Trilinear coordinates of Hofstadter zero-point
 = lim "r" → 0 ( sin "rA" / sin ( "A" − "rA") , sin "rB" / sin ( "B" − "rB" ) , sin "rC" / sin ( "C" − "rC") ) = lim "r" → 0 ( sin "rA" / "r" sin ( "A" − "rA") , sin "rB" / "r" sin ( "B" − "rB" ) , sin "rC" / "r" sin ( "C" − "rC") )
 = lim "r" → 0 ( "A" sin "rA" / "rA" sin ( "A" − "rA") , "B" sin "rB" / "rB" sin ( "B" − "rB" ) , "C" sin "rC" / "rC" sin ( "C" − "rC") )
 = ( "A" / sin "A" , "B" / sin "B" , "C" / sin "C" ) ), as lim "r" → 0 sin "rA" / "rA" = 1, etc.
 = ( "A" / "a", "B" / "b", "C" / "c" )
Trilinear coordinates of Hofstadter one-point
 = lim "r" → 1 ( sin "rA" / sin ( "A" − "rA") , sin "rB" / sin ( "B" − "rB" ) , sin "rC" / sin ( "C" − "rC") )
 = lim "r" → 1 ( ( 1 − "r" ) sin "rA" / sin ( "A" − "rA") , ( 1 - "r" ) sin "rB" / sin ( "B" − "rB" ) , ( 1 − "r" )sin "rC" / sin ( "C" − "rC") )
 = lim "r" → 1 ( ( 1 − "r" ) "A" sin "rA" / "A" sin ( "A" − "rA") , ( 1 − "r" ) "B" sin "rB" / "B" sin ( "B" − "rB" ) , ( 1 − "r" ) "C" sin "rC" / "C" sin ( "C" − "rC") ) = ( sin "A" / "A" , sin "B" / "B" , sin "C" / "C" ) ) as lim "r" → 1 ( 1 − "r" ) "A" / sin ( "A" − "rA" ) = 1, etc.
 = ( "a" / "A", "b" / "B", "c" / "C" )
Hofstadter triangles
In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers.