• Netay, Igor V. (Institute for Information Transmission Problems, RAS) ;
  • Savvateev, Alexei V. (Dmitry Pozharsky University Moscow Institute of Physics and Technology New Economic School)
  • Received : 2016.08.17
  • Accepted : 2017.02.24
  • Published : 2017.09.30


The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are parametrized by an open subset of an elliptic curve. Also we prove that there are infinitely many non-similar integer Sharygin triangles.


Supported by : Russian Foundation for Sciences, Ministry of Education and Science of the Russian Federation


  1. L. Bankoff and J. Garfunkel, The Heptagonal Triangle, Math. Mag. 46 (1973), no. 1, 7-19.
  2. C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over $\mathbb{Q}$ : wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843-939.
  3. B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225-320.
  4. R. Hartshorne, Algebraic Geometry, Springer-Verlag, New-York, 1977.
  5. H. Hasse, Beweis des Analogons der Riemannschen Vermutung fur die Artinschen und F. K. Schmidtschen Kongruenzzetafunktionen in gewissen eliptischen Fallen. Vorlaufige Mitteilung, Nachr. Ges. Wiss. Gottingen I, Math.-Phys. Kl. Fachgr. I Math. Nr. 42 (1933), 253-262.
  6. H. Hasse, Abstrakte begrundung der komplexen multiplikation und riemannsche vermutung in funktionenkorpen, Abh. Math. Sem. Univ. Hamburg 10 (1934), no. 1, 325-348.
  7. V. A. Kolyvagin, Finiteness of E($\mathbb{Q}$) and X(E;$\mathbb{Q}$) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522-540.
  8. V. A. Kolyvagin, Euler systems, in The Grothendieck Festschrift, Vol. II, 435-483, Progr. Math., 87, Birkhauser Boston, Boston, MA, 1990.
  9. E. Lutz, Sur l'equation $y^2=x^3-Ax-B$ dans les corps p-adic, J. Reine Angew. Math. 177 (1937), 238-247.
  10. S. Markelov, Diophantine... bisectors!, unpublished manuscript accepted to Kvant, 2017.
  11. T. Nagell, Solution de quelque problemes dans la theorie arithmetique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I (1935), no. 1, 1-25.
  12. K. Rubin and A. Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soc. 39 (2002), no. 4, 455-474.
  13. I. F. Sharygin, About bisectors, J. Kvant 1983 (1983), no. 8, 32-36.
  14. I. F. Sharygin, Problems in Geometry (Planimetry), Nauka, 1982.
  15. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 2000.
  16. T. H. Skolem, Diophantische Gleichungen, Chelsea, 1950.
  17. J. T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179-206.
  18. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), no. 3, 553-572.
  19. S. Tokarev, Problem M2001, Kvant 3 (2006), 17.
  20. A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), no. 3, 443-551.