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.

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 *h _{a}*, the median

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 *I _{a}*,

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.

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, *H _{a}H_{b}H_{c}* its orthic
triangle, and

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

*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*.

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.

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.