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Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721

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Marius Coman, researcher, Bucharest, Romania. Prof. Valeri Kroumov, Okayama University of Science, Japan. Said Broumi, University of Hassan II Mohammedia, Casablanca, Morocco. Dr. Ştefan Vlăduţescu, University of Craiova, Romania.

Many books can be downloaded from the following 'LJLWDO/LEUDU\RI6FLHQFH: http://fs.gallup.unm.edu/eBooks-otherformats.htm ($1 ,6%1

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This book contains 21 papers of plane geometry. It deals with various topics, such as: quasi-isogonal cevians, nedians, polar of a point with respect to a circle, anti-bisector, aalsonti-symmedian, anti-height and their isogonal. A nedian is a line segment that has its origin in a triangle’s vertex and divides the opposite side in Q equal segments. The papers also study distances between remarkable points in the 2D-geometry, the circumscribed octagon and the inscribable octagon, the circles adjointly ex-inscribed associated to a triangle, and several classical results such as: Carnot circles, Euler’s line, Desargues theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s theorem, Pantazi’s theorem, and Newton’s theorem. Special attention is given in this book to RUWKRORJLFDO WULDQJOHV ELRUWKRORJLFDO WULDQJOHV RUWKRKRPRORJLFDOWULDQJOHVand WULKRPRORJLFDOWULDQJOHV The notion of “ortho-homological triangles” was introduced by the Belgium mathematician Joseph Neuberg in 1922 in the journal Mathesis and it characterizes the triangles that are simultaneously orthogonal (i.e. the sides of one triangle are perpendicular to the sides of the other triangle) and homological. We call this “ortho-homological of first type” in order to distinguish it from our next notation. In our articles, we gave the same denomination “ortho-homological triangles” to triangles that are simultaneously orthological and homological. We call it “ortho-homological of second type.” Each paper is independent of the others. Yet, papers on the same or similar topics are listed together one after the other. This book is a continuation of the previous book 7KH*HRPHWU\RI+RPRORJLFDO7ULDQJOHV, by Florentin Smarandache and Ion Pătraşcu, Educ. Publ., Ohio, USA, 244 p., 2012. The book is intended for College and University students and instructors that prepare for mathematical competitions such as National and International Mathematical Olympiads, or the AMATYC (American Mathematical Association for Two Year Colleges) student competition, or Putnam competition, Gheorghe Ţiteica Romanian student competition, and so on.

The book is also useful for geometrical researchers.

The authors

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Ion Pătraşcu – National College “Fraţii Buzeşti, Craiova, Romania Florentin Smarandache – New Mexico University, U.S.A.

In [1] we proved, using barycentric coordinates, the following theorem:

7KHRUHP(generalization of the C. Coşniţă theorem) If 3 is a point in the triangle’s $%& plane, which is not on the circumscribed triangle,

$%&''' is its pedal triangle and $%&111,, three points such that ))))&)))& ))))&)))& ))))&))))& * 3$'' 3$11 3% 3% 3& ', 3& 1 N N 5 , then the lines $$11, %% , && 1 are concurrent. Bellow, will prove, using this theorem, the following:

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If the triangles $%& and $%&111 are orthological and their orthological centers coincide, then the lines $$11, %% , && 1 are concurrent (the triangles $%& and $%&111 are homological). 3URRI

Let 2 be the unique orthological center of the triangles $%& and $%&111 and

C1 X1 B1 Y2 Y3 O X2 B X3 Y1 C

^`;$2%&11 1

^`;%2$&21 1

^`;&2$%31 1 We denote

^<2$%&11`

^`<2%$&21

^`<2&$%31

We observe that 2$<311 2& ; (angles with perpendicular sides). Therefore: 2< sin 2$< 3 3 2$ , 2;1 sin 2&11 ; 2&1 then

2;131 2$ 2< 2& (1) Also

2&12 ; 2%< 3 therefore

2; 2 sin 2&12 ; 2&1 2< sin 2%< 3 3 2% and consequently:

2;231 2% 2< 2& (2) Following the same path:

2; 212< sin 2$12 ; sin 2%<1 2&1 2% from which

2;211 2% 2$ 2< (3) Finally

2; 3 2<1 sin 2$13 ; sin 2&<1 2$1 2& from which:

2;311 2& 2$ 2< (4) The relations (1), (2), (3), (4) lead to

2;123 2$ 2; 2% 2; 2& (5)

From (5) using the Coşniţă’s generalized theorem, it results that $$11, %% , && 1 are concurrent.

2EVHUYDWLRQ

If we denote 3 the homology center of the triangles $%& and $%&111 and G is the intersection of their homology axes, them in conformity with the Sondat’s theorem, it results that 23A G .

5HIHUHQFHV [1] Ion Pătraşcu – Generalizarea teoremei lui Coşniţă – Recreaţii Matematice, An XII, nr. 2/2010, Iaşi, Romania [2] Florentin Smarandache – Multispace & Multistructure, Neutrosophic Transdisciliniarity, 100 Collected papers of science, Vol. IV, 800 p., North- European Scientific Publishers, Honka, Finland, 2010.

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Prof. Claudiu Coandă – National College “Carol I”, Craiova, Romania Prof. Florentin Smarandache – University of New Mexico, Gallup, U.S.A. Prof. Ion Pătrașcu – National College “Fraţii Buzeşti”, Craiova, Romania

In this article we prove a theorem that will generalize the concurrence theorems that are leading to the Franke’s point, Kariya’s point, and to other remarkable points from the triangle geometry.

7KHRUHP Let 3 D,,EJ and $ ',%& ', ' its projections on the sides %&&$, respectively $% of the $%& triangle . JJJJG JJJG JJJJG JJJG JJJJG JJJJG We consider the points $", %& ", " such that 3$N3$"' , 3%N3%"' , 3&N3&"' , where N5 * . Also we suppose that $$%%&&', ', ' are concurrent. Then the lines $$%%&&", ", " are concurrent if and only if are satisfied simultaneously the following conditions: §·§·§·ED J E DJ DEF$¨¸¨¸¨¸cos cos %D%&E&$EJ cos cos JD cos cos 0 ©¹©¹©¹ED FE DF DJE222§·§·§· EDJ J ED cos$%¨¸¨¸¨¸ cos cos & cos %& cos cos $ cos & cos $ cos % 0 DFE222©¹©¹©¹ EDF FED 3URRI We find that

§·DD222 222 $'0,¨¸ DEF E, DEF J ©¹22DD22 JJJJG JJJG JG JG JJG ªºDD222 222 3$N3$NUDEFUDEFU"' D $%& ¬¼«»22DD22 JJJJGJGJGJJG 3$"" DD U$%& EE " U JJ " U We have: °DD" N D ° ° ND EE DEF222 ® " 2 , ° 2D ° ND JJ DEF222 ° " 2 ¯ 2D Therefore:

ϲϵ °DD"1 N ° ° ND E DEF222E ® " 2 ° 2D ° ND J DEF222J ° " 2 ¯ 2D Hence:

§·NNDD222 222 $"¨¸ 1 ND , DEFEJ, DEF ©¹22DD22 Similarly:

§·EE222 222 %DEFDEF',¨¸ D 0, J ©¹22EE22

§·NNEE222 222 %DEFN",¨¸ D 1-EJ, DEF ©¹22EE22

§·JJ222 222 &',¨¸ DEFDE DEF , 0 ©¹22FF22

§·NNJJ222 222 &",¨¸ DEFD DEF EJ, 1- N ©¹22FF22 Because $$%%&&', ', ' are concurrent, we have:

D 222 EJ222 222 222 DEF EJD DEF DEF 222DEF 1 DE J DEF222JDE DEF222 DEF222 22DE22 2 F 2 We note D D 0 DEF222 Ecos & 2DD2 DD 1DEFF% 222 cos 2DD2 E E 3 DEF222 Fcos $ 2EE2 EE 4DEFD& 222 cos 2EE2 J J 5 DEF222 Dcos % 2FF2 JJ 6 DEF222 Dcos $ 2FF2 The precedent relation becomes 035E JD 1 146JDE The coefficients 0 ,,,,,13456 verify the following relations:

ϳϬ 01 D 34 E 56 J D 0ED 2 2 D 4DE 2 E E2 E 3FE 2 2 E 6EJ 2 J F2 J 5DJ 2 2 F 1FD 2 D D2 035 Therefore 1 461 0 E35 J D DEJ DE 350033550035 EJ JD D E J 1JDEDEJDEEJJDDEJ 4 6 1 4 6 16 14 46 146 . We deduct that: DEEJJDD3 5 0 03 E 35 J 50 DEEJJDD 1 4 6 16 E 14 J 46 146 (1) We apply the theorem:

Given the points 4DEFLLLL(,,), L 1,3 in the plane of the triangle $%& , the lines

EF12D3 $4123,, %4 &4 are concurrent if and only if 1. FDE123 For the lines $$%%&&", ", " we obtain N0E N3DD N5 1. N1JEE N6 N6 It result that N350N0335502 DE EJ JD D E J N146N1614462 DE EJ JD D E J (2) For relation (1) to imply relation (2) it is necessary that DEEJJDDEEJJD350 146 and D 16EJDEJ 14 46 03 35 50 or

ϳϭ §·§·§·ED J E DJ °DEF$¨¸¨¸¨¸cos cos %D%&E&$EJ cos cos JD cos cos 0 ° ©¹©¹©¹ED FE DF ® 222 °DJE§·§·§· EJE J ED cos$%¨¸¨¸¨¸ cos cos & cos %% cos cos & cos & cos $ cos % 0 ¯° DFE222©¹©¹©¹ EFE FED As an open problem, we need to determine the set of the points from the plane of the triangle $%& that verify the precedent relations. We will show that the points , and 2 verify these relations, proving two theorems that lead to Kariya’s point and Franke’s point.

7KHRUHP (Kariya -1904) Let , be the center of the circumscribe circle to triangle $%& and $ ',%& ', ' its projections on the sides %&&$$%, , . We consider the points $",%& ", " such that: JJJG JJJG JJJG JJJG JJJG JJJJG ,$" N ,$ ', ,% " N ,% ', ,& " N ,& ', N 5* . Then $$%%&&", ", " are concurrent (the Kariya’s point) 3URRI §·DEF The barycentric coordinates of the point , are , ¨¸,, . ©¹222S SS Evidently: DEF cos $ cos % DEF cos % cos & DEF cos & cos $ 0 and cos$% cos cos & cos %& cos cos $ cos &$ cos cos % 0 . In conclusion $$%%&&", ", " are concurrent.

7KHRUHP (de Boutin - 1890) Let 2 be the center of the circumscribed circle to the triangle $%& and $ ',%& ', ' its projections on the sides %&&$$%, , . Consider the points $", %& ", " such that 2$''' 2% 2& NN5, * . Then the lines $$%%&&", ", " are concurrent (The point of 2$""" 2% 2& Franke – 1904). 3URRI §·555222 sin 2%$ cos sin 2 $% cos 2$%&¨¸sin 2 , sin 2 , sin 2 , 31 , because 0 . ©¹222666 sin% sin $ Similarly we find that 5 4 and 0 6 . Also D 031635145046 DE, E, J J. It is also verified the second relation from the theorem hypothesis. Therefore the lines $$%%&&", ", "are concurrent in a point called the Franke’s point.

5HPDUN: It is possible to prove that the Franke’s points belong to Euler’s line of the triangle $%& .

7KHRUHP:

ϳϮ Let ,D be the center of the circumscribed circle to the triangle $%& (tangent to the side %& ) and $ ', %& ', ' its projections on the sites %&&$$%, , . We consider the points JJJG JJJG JJJG JJJG JJJG JJJJG $", %& ", " such that ,$" N ,$ ', ,% " N ,% ', ,& " N ,& ', N 5* . Then the lines $$%%&&", ", "are concurrent. 3URRI §·DE F , D ¨¸,, ; ©¹222 S DSDSD The first condition becomes: DEF cos $ cos % DEF cos % cos & DEF cos & cos $ 0 , and the second condition: cos$% cos cos & cos % cos & cos $ cos &$ cos cos % 0

Is also verified. From this theorem it results that the lines $$%%&&", ", "are concurrent.

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Similarly, this theorem is proven for the case of ,E and ,F as centers of the ex-inscribed circles.

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[1] C. Coandă –Geometrie analitică în coordonate baricentrice – Editura Reprograph, Craiova, 2005. [2] F. Smarandache – Multispace & Multistructure, Neutrosophic Trandisciplinarity (100 Collected Papers of Sciences), Vol. IV, 800 p., North-European Scientific Publishers, Finland, 2010.

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