A. A. Buturlakin, S. S. Presnyakov, D. O. Revin, S. A. Savin, “Area of a triangle and angle bisectors”, Sib. Èlektron. Mat. Izv., 2020, Volume 17,Pages <nobr>732

Geometry and topology

Area of a triangle and angle bisectors

A. A. Buturlakin^ab, S. S. Presnyakov^c, D. O. Revin^ba, S. A. Savin^dc

^a Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
^b Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia
^c Specialized Educational Scientific Center of Novosibirsk State University, 1/1, Pirogova str., Novosibirsk, 630090, Russia
^d The Orthodox Gymnasium in the name saint Sergius of Radonezh, 3, Akademicheskaya str., Novosibirsk, 630090, Russia

Abstract: Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to express the area of $ABC$ in radicals of $l_a,l_b,l_c$.

Keywords: area of a triangle, angle bisectors, ruler and compass construction, Galois group of a polynomial, algebraic equation, solution in radicals.

UDC: 514.112.3, 512.622

MSC: 51M04, 12F10

Received May 6, 2020, published May 31, 2020

Language: English

DOI: 10.33048/semi.2020.17.052