## Volume 4

1.

*Robert A. Russell*,
Inscribed Equilateral Triangles.

Abstract
Article
*If equilateral triangles are inscribed in an irregular triangle in such a way that each edge (or its extension) of the latter contains a distinct vertex of each equilateral triangle, the centers of the equilateral triangles must lie on one of two lines, each of which is orthogonal to the Euler line of the irregular triangle.
*

2.

*Dasari Naga Vijay Krishna*,
A new proof of Ptolemy's Theorem.

Abstract
Article
*In this article we give a new proof of well-known Ptolemy’s Theorem of a Cyclic Quadrilaterals.*

3.

*Tran Quang Hung*,
Some extensions of the Droz-Farny line theorem.

Abstract
Article
*We provide some generalizations of the Droz-Farny theorem and give synthetics proofs for them.*

4.

*Tran Thu Le and Kien Trung Nguyen*,
The Five Conics Problem.

Abstract
Article
*This paper concerns a generalization of the so-called three conics theorem. We consider the model that consists of five conics. We show that the five conics
in the setting has the same property as the three conics theorem and it takes the three conics in the model as a special case. Interesting corollaries are discussed as well.*

5.

*Nguyen Minh Ha and Pham Nam Khanh*,
Another simple proof of the Goormaghtigh theorem and the generalized Goormaghtigh
theorem.

Abstract
Article
*We introduce a simple proof of the Goormaghtigh theorem and the
generalized Goormaghtigh theorem using the concept of cross ratio.*

6.

*Ismail M. Isaev, Yuri N. Maltsev, and Anna S. Monastyreva*,
On Some Geometric Relations of a Triangle.

Abstract
Article
*
For a triangle
$\mathrm{ABC}$ we consider the circles passing through a vertex of the triangle and tangent to the oppisite side as well as to the circumcircle.
We prove that
$\genfrac{}{}{0.1ex}{}{1}{{r}_{a}}}+{\displaystyle \genfrac{}{}{0.1ex}{}{1}{{r}_{b}}}+{\displaystyle \genfrac{}{}{0.1ex}{}{1}{{r}_{c}}}={\displaystyle \genfrac{}{}{0.1ex}{}{2}{R}}+{\displaystyle \genfrac{}{}{0.1ex}{}{1}{r}$, where
${r}_{a}$,
${r}_{b}$,
${r}_{c}$ are the radii of the these three circles, and
$R$,
$r$ are the circumradius and the inradius of the triangle
$\mathrm{ABC}$,
respectively. This equation generalizes the main result from [1].
*

7.

*Hiroshi Okumura*,
The arbelos in Wasan geometry, problems of Izumiya and Naitō.

Abstract
Article
*We generalize two sangaku problems involving an arbelos proposed by Izumiya and Naitō, and show the existence of six non-Archimedean congruent circles.*

8.

*Paris Pamfilos*,
Parabolas and families of convex quadrangles.

Abstract
Article
*In this article we study some parabolas naturally associated with a generic convex quadrangle. It is shown that the quadrangle defines, through these parabolas, a one-parameter-family of quadrangles, containing the quadrangle of reference. The members of this family share the same diagonal lines, Newton line and ratio of diagonals. In addition it is shown that this family contains a unique cyclic quadrangle.*

XI Geometrical Olympiad in Honour of I. F. Sharygin.

The Correspondence Round,
pdf

XII Geometrical Olympiad in Honour of I. F. Sharygin.

The Correspondence Round,
pdf

## Volume 3

1.

*Artemy A. Sokolov and Maxim D. Uriev*,
On Brocard's points in polygons, pp. 1-3.

Abstract
Article
*In this note we present a synthetic proof of the key lemma, defines in the problem of A A. Zaslavsky.*

2.

*Pavel E. Dolgirev*,
On some properties of confocal conics, pp. 4-11.

Abstract
Article
*We prove two theorems concerning confocal conics. The first one is related to bodies invisible from one point. In particular, this theorem is a generalization of Galperin—Plakhov's theorem. *

The second one is related to billiards bounded by confocal conics and is used to construct bodies invisible from two points. All the proofs are synthetic.

3.

*Paris Pamfilos*,
Ellipse generation related to orthopoles, pp. 12-34.

Abstract
Article
*In this article we study the generation of an ellipse related to two intersecting circles. *

The resulting configuration has strong ties to triangle geometry and by means of orthopoles establishes also a relation with cardioids and deltoids.

4.

*Danylo Khilko*,
Some properties of intersection points of Euler line and orthotriangle, pp. 35-42.

Abstract
Article
*We consider the points where the Euler line of a given triangle **ABC* meets the sides of its orthotriangle, i.e. the triangle whose vertices are feet of the altitudes of *ABC*.
In this note we study properties of these points and how they relate to the known objects.

5.

*Pavel A. Kozhevnikov and Alexey A. Zaslavsky*,
On Generalized Brocard Ellipse, pp. 43-52.

Abstract
Article
*
For a fixed point **P* and a fixed circle *Ω* consider a conic (the generalized Brocard ellipse) that touches lines *XY*
inclined at a fixed angle to *PX*, where *X∈Ω*.
For this construction, we prove some facts that allow to obtain more properties of harmonic quadrilaterals.

6. Problem section, pp. 53-55.

pdf
7. Geometrical Olympiad in Honour of I. F. Sharygin.

The Correspondence Round, pp. 56-59.

pdf
The Final Round, pp. 60-62.

pdf
Download the whole volume:

pdf.

1.

*Dimitar Belev*,
Some Properties of the Brocard Points of a Cyclic Quadrilateral, pp. 1—10.

Abstract
Article
*In this article we have constructed the Brocard points of a cyclic quadrilateral, we have found some of their properties and using these properties we have proved the problem of A. A. Zaslavsky.*

2.

*Nikolai Ivanov Beluhov*,
A Curious Geometric Transformation, pp. 11—25.

Abstract
Article
*An expansion is a little-known geometric transformation which maps directed circles to directed circles. We explore the applications of expansion to the solution of various problems in geometry by elementary means.*

3.

*Debdyuti Banerjee and Sayan Mukherjee*,
Neuberg Locus And Its Properties, pp. 26—38.

Abstract
Article
*In this article we discuss the famous Neuberg Locus.
We also explore some special properties of the cubic, and provide purely synthetic proofs to them.*

4.

*Tran Quang Hung and Pham Huy Hoang*,
Generalization of a problem with isogonal conjugate points, pp. 39—42.

Abstract
Article
*In this note we give a generalization of the problem that was used in the All-Russian Mathematical Olympiad and a purely sythetic proofs.*

5.

*Ilya I. Bogdanov, Fedor A. Ivlev, and Pavel A. Kozhevnikov*,
On Circles Touching the Incircle, pp. 43—52.

Abstract
Article
*For a given triangle, we deal with the circles tangent to the incircle and passing through two its vertices. We present some known and recent properties of the points of tangency and some related objects.*

Further we outline some generalizations for polygons and polytopes.

6.

*Alexey A. Zaslavsky*,
One property of the Jerabek hyperbola and its corollaries, pp. 53—56.

Abstract
Article
*We study the locus of the points P having the following property: if
A*_{1}B_{1}C_{1}
is the circumcevian triangle of P with respect to the given triangle ABC, and
A_{2},
B_{2},
C_{2}
are the reflections of A_{1}, B_{1}, C_{1} in BC, CA, AB, respectively, then the triangles ABC and A_{2}B_{2}C_{2} are perspective.
We show that this locus consists of the infinite line and the Jerabek hyperbola of ABC. This fact yields some interesting corollaries.

7.

*Fedor K. Nilov*,
A generalization of the Dandelin theorem, pp. 57—65.

Abstract
Article
*We prove three apparently new theorems related to the doubly tangent circles of conics including a generalization of the Dandelin theorem on spheres inscribed in a cone. Also we discuss the focal properties of doubly tangent circles of conics.*

8.

*Alexander Skutin*,
On rotation of a isogonal point, pp. 66—67.

Abstract
Article
*In this short note we give a synthetic proof of the problem posed by A. V Akopyan in [1].
We prove that if Poncelet rotation of triangle T between circle and ellipse is given then the locus of the isogonal conjugate point of any fixed point P with respect to T is a circle.*

9. Problem section, pp. 68—69.

pdf
10. IX Geometrical Olympiad in Honour of I. F. Sharygin.

The Correspondence Round, pp. 70—72.

pdf
The Final Round, pp. 73—76.

pdf
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pdf.

1.

*Ilya I. Bogdanov*, Two theorems on the focus-sharing ellipses: a three-dimensional view, pp. 1—5.

Abstract
Article
*Consider three ellipses each two of which share a common focus.
The “radical axes” of the pairs of these ellipses are concurrent, and the points of
intersection of the common tangents to the pairs of these ellipses are
collinear.*

We present short synthetical proofs of these facts. Both proofs deal with the
prolate spheroids having the given ellipses as axial sections.

2.

*Alexey A. Pakharev*, On certain transformations preserving perspectivity of triangles, pp. 6—16.

Abstract
Article
*
To any pair of perspective triangles we assign a family F of projective transformations such that image of the second
triangle under any transformation from F is perspective to the first triangle. This helps us to solve some interesting
problems.
*

3.

*Lev A. Emelyanov and Pavel A. Kozhevnikov*, Isotomic similarity, pp. 17—22.

Abstract
Article
*
Let A*_{1}, B_{1}, C_{1} be points chosen on the sidelines BC, CA, BA of a triangle ABC, respectively. The circumcircles
of triangles AB_{1}C_{1}, BC_{1}A_{1}, CA_{1}B_{1} intersect the circumcircle of triangle ABC again at points A_{2}, B_{2}, C_{2}
respectively. We prove that triangle A_{2}B_{2}C_{2} is similar to triangle A_{3}B_{3}C_{3}, where A_{3}, B_{3}, C_{3} are symmetric to A_{1},
B_{1},C_{1} with respect to the midpoints of the sides BC, CA, BA respectively.

4.

*Arseniy V. Akopyan*, Conjugation of lines with respect to a triangle, pp. 23—31.

Abstract
Article
*
Isotomic and isogonal conjugate with respect to a triangle is a well-known and well studied map frequently used in classical geometry. In this article we show that there is a reason to study conjugation of lines. This conjugation has many interesting properties and relations to other objects of a triangle.
*

5.

*Nguyen Minh Ha*, A proof of Vittas’ Theorem and its converse, pp. 32—39.

Abstract
Article
*
We discuss Vittas’s theorem, which states that the Euler lines of non-equilateral triangles ABP, BCP,
CDP and DAP in a cyclic quadrilateral ABCD, whose diagonals AC and BD intersect at P, are concurrent or are pairwise parallel or coincident. We also introduce and prove the converse of these theorems.
*

6.

*Darij Grinberg*, Ehrmann’s third Lemoine circle, pp. 40—52.

Abstract
Article
*
The symmedian point of a triangle is known to give rise to two circles, obtained by drawing respectively parallels and antiparallels to the sides of the triangle through the symmedian point. In this note we will explore a third circle with a similar construction — discovered by Jean-Pierre Ehrmann. It is obtained by drawing circles through the symmedian point and two vertices of the triangle, and intersecting these circles with the triangle’s sides. We prove the existence of this circle and identify its center and radius.
*

7.

*Nikolai Ivanov Beluhov*, An elementary proof of Lester’s theorem, pp. 53—56.

Abstract
Article
*
In 1996, J. A. Lester discovered that in every scalene triangle the two Fermat-Torricelli points, the circumcenter, and the center of the nine-point circle are concyclic. We give the first proof of this fact to only employ results from elementary geometry.
*

8.

*Dmitry S. Babichev*, Circles touching sides and the circumcircle for inscribed quadrilaterals, pp. 57—61.

Abstract
Article
*
In an inscribed quadrilateral, four circles touching the circumcircle and two neighboring sides have a radical center.
*

9.

*Vladimir N. Dubrovsky*, Two applications of a lemma on intersecting circles, pp. 62—64.

Abstract
Article
*
A useful property of the direct similitude that maps one of two intersecting circles on another and fixes their common point is applied to the configuration consisting of a triangle, its circumcircle, and a circle through its vertex and the feet of its two cevians.
*

10.

*Alexey A. Zaslavsky*, Geometry of Kiepert and Grinberg–Myakishev hyperbolas, pp. 65—71.

Abstract
Article
*
A new synthetic proof of the following fact is given: if three points A′, B′, C′ are the apices of
isosceles directly-similar triangles BCA′, CAB′, ABC′ erected on the sides BC, CA, AB of a triangle
ABC, then the lines AA′, BB′, CC′ concur. Also we prove some interesting properties of the Kiepert hyperbola which is the locus of concurrence points, and of the Grinberg–Myakishev hyperbola which is its generalization.
*

11. Problem section, pp. 72—74.

pdf
12. Geometrical olympiad in honor of I.F. Sharygin, pp. 75—86.

pdf
Download the whole volume:

pdf.