Lexicographical account of constructional problems of triangle geometry problems

Source : Mathematicals in schools, 1937 (5) pp. 4–30, 1937 (6) pp. 21–45, Moscow, USSR.

Abstract : A standard problem in geometry is to construct a triangle with given elements (such as side, angles, medians, bisectors, altitudes, inradius, exradii, circumradius, half-perimeter, area) ; generally you need three elements. Sometimes this construction can be performed by a compassand straightedge, sometimes not. The paper gives a complete list of all triples that allows such a construction, but not proofs for the cases when author claims it does not exist (though some methods used for one of the cases are mentioned).

For some problems the restriction on input data (needed to ensure that triangle in question exists) are given.

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Un problème de construction
Об одной задаче построения

Abstract : This paper answers a question from Luís Lopes to group Hyacinthos in his message 4440, Wednesday, November 28, 2001 ("german" triangle construction), on the construction of a triangle, given the altitude ha, the median mb, the internal bissector ba.

In the article " Лексикографическое изложение конструктивных задач геометрии треугольника " by V. B. Fursenko, this problem appears under the nummer 232, with the mention "unsolvable". It appears also in the book Die Konstruktion von Dreiecken by Herterich under the nummer 179.

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Sur quelques propriétés des triangles podaires des centres des cercles exinscrits

Abstract : This paper answers a question from Nikolaos Dergiades to group Hyacinthos in his message 4826, Friday, February 22th, 2002 (The pedal triangles of Ia, Ib, Ic). We studie in this note some properties of a few triangle centers with respect to these pedal triangles.

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The center of the Taylor circle

Abstract : This paper answers a question from Paul Yiu to group Hyacinthos. How give a synthetic proof that the center of the Taylor circle of a given triangle is the Spieker center of its orthic triangle ? More properties of the orthic triangle, its medial triangle and the Taylor circle can be found, with synthetic proofs, in Yvonne et René SORTAIS, La Géométrie du triangle, HERMANN, Paris, 1987, pp. 26–41.

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Une application affine sur les hauteurs d’un triangle

Abstract : This paper gives a synthetic proof of the results expressed by Antreas P. Hatzipolakis in his message 5050, Friday, March 22th, 2002 (Eulerline), to group Hyacinthos. Let ABC be a triangle, HaHbHc its orthic triangle, and t a parameter, we study triangles in perspective, defined from the points Ma, Mb and Mc on the altitudes such that: AMa = t.AHa, BMa = t.BHa and CMa = t.CHa.

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Cubiques invariantes par transformation du second ordre

(Draft version)

Abstract : This paper presents some properties of isocubics. I would appreciate all kinds of suggestions or improvements. Please sent a mail

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The Napoleon configuration

in Forum Geometricorum , 2 (2002) pp. 39 – 46

Abstract : It is an elementary fact in triangle geometry that the two Napoleon triangles are equilateral and have the same centroid as the reference triangle. We recall some basic properties of the Fermat and Napoleon configurations, and use them to study equilateral triangles bounded by cevians. There are two families of such triangles, the triangles in the two families being oppositely oriented. The locus of the circumcenters of the triangles in each family is one of the two Napoleon circles, and the circumcircles of each family envelope a conchoid of a circle.

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Étude sur les droites de Steiner

Abstract : This paper gives a synthetic proof of the results conjectured by Antreas P. Hatzipolakis in his message 5336, Wednesday, May 1st, 2002 (Euler Line [Conjectures]), to group Hyacinthos.

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Huit triades de cercles de même rayon

en collaboration avec Edward BRISSE

Abstract : We study in this note a triad of circles, centered on the internal angle bisector of a triangle ABC, with the same radius, and tangent each one to two sidelines of ABC. We prove that there exists only two triades of concurrent such circles, the points of concurrence are both similitude center of the circumcircle and the incircle.

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Une propriété isotomique des droites passant par le centre de gravité

Abstract : This paper gives a synthetic proof of the result expressed by Bernard Gibert in his message 5848, Wednesday, August 14th, 2002 (a question), to group Hyacinthos. Let ABC be a triangle, M any point, N its reflection about G (centroid of ABC), M* and N* their isotomic conjugates, then the lines MN and M*N* are parallel. We also prove the lines MN* and NM* are intersecting at a point lying on the Steiner circumellipse.

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