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Variance on Topics of Plane Geometry $5 251251 \\ 3 &[ 'D(DD ( D % VARIANCE ON TOPICS OF PLANE GEOMETRY ,RQ3ăWUDúFX)ORUHQWLQ6PDUDQGDFKH 9$5,$1&(21723,&6 2)3/$1(*(20(75< (GXFDWLRQDO3XEOLVKLQJ 2013 1 Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 &RS\ULJKWE\3XEOLVKHUDQG$XWKRUV 3HHU5HYLHZHUV 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 2 &217(176 9$5,$1&(21723,&62)3/$1(*(20(75< 3UHIDFH 4XDVL,VRJRQDO&HYLDQV 1HGLDQVDQG7ULDQJOHVZLWKWKH6DPH&RHIILFLHQWRI'HIRUPDWLRQ )URPD3UREOHPRI*HRPHWULFDO&RQVWUXFWLRQWRWKH&DUQRW&LUFOHV 7KH3RODURID3RLQWZLWK5HVSHFWWRD&LUFOH 6HYHUDO0HWULFDO5HODWLRQV5HJDUGLQJWKH$QWL%LVHFWRUWKH$QWL6\PPHGLDQWKH$QWL +HLJKWDQGWKHLU,VRJRQDO $Q ,PSRUWDQW $SSOLFDWLRQ RI WKH &RPSXWDWLRQ RI WKH 'LVWDQFHV EHWZHHQ 5HPDUNDEOH 3RLQWVLQWKH7ULDQJOH*HRPHWU\ 7KH'XDOLW\DQGWKH(XOHU¶V/LQH 7ZR$SSOLFDWLRQVRI'HVDUJXHV¶7KHRUHP $Q$SSOLFDWLRQRI6RQGDW¶V7KHRUHP5HJDUGLQJWKH2UWKRKRPRORJLFDO7ULDQJOHV $QRWKHU3URRIRIWKHD7KHRUHP5HODWLYHWRWKH2UWKRORJLFDO7ULDQJOHV 7ZR 7ULDQJOHV ZLWK WKH 6DPH 2UWKRFHQWHU DQG D 9HFWRU 3URRI RI 6WHYDQRYLF¶V 7KHRUHP 7ZR5HPDUNDEOH2UWKR+RPRORJLFDO7ULDQJOHV $*HQHUDOL]DWLRQRI&HUWDLQ5HPDUNDEOH3RLQWVRIWKH7ULDQJOH*HRPHWU\ *HQHUDOL]DWLRQRID5HPDUNDEOH7KHRUHP 3DQWD]L¶V7KHRUHP5HJDUGLQJWKH%L2UWKRORJLFDO7ULDQJOHV $1HZ3URRIDQGDQ$SSOLFDWLRQRI'HUJLDGHV¶7KHRUHP 0L[W/LQHDU&LUFOHV$GMRLQWO\([,QVFULEHG$VVRFLDWHGWRD7ULDQJOH 3 $3URSHUW\RIWKH&LUFXPVFULEHG2FWDJRQ )URP1HZWRQ¶V7KHRUHPWRD7KHRUHPRIWKH,QVFULEDEOH2FWDJRQ 7ULSOHWVRI7UL+RPRORJLFDO7ULDQJOHV $&ODVVRI2UWKR+RPRORJLFDO7ULDQJOHV 4 3UHIDFH 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. 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