International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM Volume 2, 2017, pp.117-134 Received 20 June 2017. Published on-line 1 July 2017 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. Computer Discovered Mathematics: Notable Circles Sava Grozdeva, Hiroshi Okumurab and Deko Dekovc 2 a VUZF University of Finance, Business and Entrepreneurship, Gusla Street 1, 1618 Sofia, Bulgaria e-mail:
[email protected] b Maebashi Gunma, 371-0123, Japan e-mail:
[email protected] cZahari Knjazheski 81, 6000 Stara Zagora, Bulgaria e-mail:
[email protected] web: http://www.ddekov.eu/ Abstract. By using the computer program ”Discoverer” we study a few notable circles of triangle. Keywords. triangle geometry, notable point, Euclidean geometry, computer dis- covered mathematics, “Discoverer”. Mathematics Subject Classification (2010). 51-04, 68T01, 68T99. 1. Introduction Let ABC be a triangle with side lengths a = BC; b = CA and c = AC. Denote by C(X; Y ) the circle having the segment joining points X and Y as its diameter. We denote by X(n) the Kimberling points [8] of triangle ABC. Hence, X(2) is the Centroid, X(3) is the Circumcenter, X(4) is the Orthocenter and X(5) is the Nine-Point Center of triangle ABC. We will study the circles C = fC(X(i);X(j)); 2 ≤ i < j ≤ 5g. Denote C(X(i);X(j) as C(i; j) for short. Note that C(2; 4) is the well known orthocentroidal circle (See [13], Orthocentroidal circle). It seems that the other circles of C are not investi- gated.