X600

3rd HATZIPOLAKIS-YIU POINT

Let OA be the circle with center A and radius R, the circumradius of triangle ABC. Let BA be the point where OA meets line AB farthest from B. Define CB and AC cyclically. Let CA be the point where OA meets line AC farthest from C. Define AB and BC cyclically. X600 is the radical center of the circles ABACA, BCBAB, CACBC. If “farthest from” is changed to “nearest to” in the construction, the resulting point is X5507. (Antreas Hatzipolakis, Paul Yiu, 1/5/01, Hyacinthos #2344, #2346-8; for a correction in coordinates, see Paul Yiu, Adgeom #202.)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_600 = 3rd HATZIPOLAKIS-YIU POINT (0.5029, 0.1275) A B C X600 (a, b, c) = (5, 6, 7) | AB = 1

X600 (3rd HATZIPOLAKIS-YIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{2}(bc + 2S)[abc(b + c - a) + 2(b^{2} + c^{2} - a^{2})S]\)
Search Value (6, 9, 13): \(4.572933349640\)
▸ Incident Central Lines1 line

{4640, 5507}