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X(1), X(2), X(10), X(81), X(191), X(366), X(1029), X(1654) harmonic associates of X(366) 

(contributed by Peter Moses) K328 is the locus of P such that the circumcenters Oa, Ob, Oc of the similar triangles {I[bc],I[cb],P}, {P,I[ca],I[ac]}, {I[ba],P,I[ab]} are perspective to ABC, where I[ab] and I[ac] are the points on the sidelines of the Aisoscelizer triangle through P. See Peter's note below. The locus of the perspector Q is pK(X1<>X1224, X1224). K328 has the same asymptotic directions as K317 = pK(X81, X86) and also K455 = pK(X2, X319). The tangents at A, B, C concur at X(81). K328 is the isogonal transform of pK(X31, X37) and the isotomic transform of pK(X75, X321). Isoscelizers : A line perpendicular to the angle bisector of vertex A is the Aisoscelizer. When intersected by the sidelines containing A, it is the base of an isosceles triangle containing vertex A. Isoscelizer is a name coined by Peter Yff in the early 1960's. Suppose the Aisoscelizer passes through P(p,q,r). This line, (c q + b r) x + (b r  c (p + r)) y +(c q  b (p + q)) z = 0, cuts the sidelines AB and AC in the points I[ab] and I[ac] with barycentrics {c (p + r)  b r, c q + b r, 0} and {b (p + q)  c q, 0, c q + b r} respectively. It also cuts the BC sideline at I[aa] = {0, b (p + q)  c q, b r  c (p + r)}, concurrent if P is on the Feuerbach hyperbola. The length of the Aisoscelizer I[ab] I[ac], passing through P, as the base of an isosceles triangle may be written as 2 (c q + b r) Sin(A / 2) / (p + q + r). If the 3 isoscelizers of a triangle have the same length and are concurrent, they meet at P = X(173), the congruent isoscelizer point. The area of the Aisoscelizer triangle, A I[ab] I[ac], is (c q + b r)^2 / (b c (p + q + r)^2) that of the area of ABC. If area of the 3 isoscelizer triangles of a triangle are equal and the isoscelizers are concurrent, they meet at P = X(364). The three triangles I[bc] I[cb] P, P I[ca] I[ac] and I[ba] P I[ab] are similar and become congruent for P = X(174). The area of I[bc] I[cb] P = (a  b  c) p^2 / (a (p + q + r)^2) that of ABC. Its circumcenter is {a p, b p + (a + b + c) q, c p + (a + b + c) r}. The Euler lines of these 3 similar triangles are concurrent for P on the line X(2) X(7). The point of concurrence is on the line X(59) X(1319). The Brocard axes of these 3 similar triangles are concurrent for P on the OI line, X(1) X(3). The point of concurrence is on the line X(36) X(59). The OI lines of these 3 similar triangles are concurrent for P on the line X(173) X(174). The circumcenters of the other 3 triangles passing through P, A I[ba] I[ca] lie on their appropriate angle bisectors, and are therefore perspective to ABC at X(1). Paul Yiu has written notes http://www.math.fau.edu/yiu/Isoscelizers.ps The figure below is drawn with P = X(366). 
