Home page  Catalogue  Classes  Tables  Glossary  Notations  Links  Bibliography  Thanks  Downloads  Related Curves 

CL003 is the class of isogonal circumstrophoids with node I, the incenter. Remember that these cubics form a pencil of focal nodal cubics. Similarly, it is possible to define three other pencils when I is replaced by an excenter. Given a point Q which is not an in/excenter, there is one and only one circumstrophoid Sq with node Q. Sq is the locus of point P such that the reflections of the cevian lines of P in the corresponding cevian lines of Q are concurrent at P', which is also a point on Sq, called the conjugate of P on Sq. All the known properties of strophoids are easily adapted in this configuration and are summarized below :


Now, if Q and R are two (distinct) isogonal conjugates, the corresponding strophoids Sq and Sr are themselves isogonal conjugates and their foci Fq, Fr are inverse in the circumcircle. Examples • with Q = X(4) and R = X(3), we obtain the Ehrmann strophoid K025 and the Jerabek strophoid K039 respectively with foci X(265) and its inverse X(5961). • with X(36) and X(80), we find the two strophoids K274 and K275. • with X(2) and X(6), we find the two strophoids K1298 and K1299. See also the Fermat strophoids K061a and K061b. Remarks — Sq is a K0 if and only if Q lies on Q039. — Sq contains O if and only if Q lies on Q023. — Sq contains H if and only if Q lies on Q038. — Sq contains I if and only if Q lies on Q181.

