Havel, Václav

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeVáclav HavelOne generalization of the fourth harmonic point (Preliminary communication)Commentationes Mathematicae Universitatis Carolinae, Vol. 8 (1967), No. 3, 463--467Persistent URL: http://dml.cz/dmlcz/105128Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1967Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCommentationes Mathematicae Univers i ta t t s Carolinae 8,3 (1967/ ONE OSNERALIZATION OP THE FOUETff H&RMONIC POINT Václav HAVEL, Brno x) Preliminary communication By a frame & in an aff ine plane & we s h a l l mean any parallelogram 0 Jx 3 J^. . The l inea 03x 7 Q3„ are ca l led coordinate a x i s . tf determines the planar ternary r ing TI (£ l ] , p . l 6 ) for which tP can be ident i f i ed with > T? x T r where 0 « (0, 0), J„* (1, 0), J * (1,1), 3^(0,1). Then to each point A e 0 3X \ { 0 } there i s exac t ly one point A'f e 0 3 x \ { 0 ) auch that A'r ** (^, 0) where a a * 4 , A - ( a , 0) * Condition ( l ) : Be given a f ixed frame F*»* 03^ 3* J * . Then for each A 6 0 3^ \ f 0 ] the point A'^ i s inde-pendent on the p o š i t ion of the variable frame ff'** 03^ 3 3 where 3L runa over 0 3^ Pronosit ion 1« In an aff ine plane & l e t there be g i -ven a f ixed frame iť* « 0 3X 3* J* . Then the conclu-s ion of ( l ) i s equivalent to the w l e f t inverse propertyH ( 2 ^ ) a, (o/Jr) m Jb> for a l l a, e T^* ^ <0}, fre 1^M where the mul t ip l i ca t ion i s taken with re spéct í o ~TLM • Convention. If the element o! with a! a, * 4 de-termined for cu € 1L^ \{0Í s a t i s f i e s a l so the eq ia t ion CL a «* 4 then we sha l l write a/ ** cc * Lemma 1 , Let T be a Veblen Wedderburn systém (Cljf , p.17) with the l e f t inverse property. Then for - 463 -(3) a f- 4 ) » - a for all a, € T , (4) f a ( - 4 ) M > » & for a l l <fc € T , i t holds (3) < = * (4), and (3) implies (5) a ( - » s - > for a l l ď ; > € T • Lemma 2 . Let a t rans la t ion affine plane ÍP sat isfy ( l ) . Then (3) holds in l^m i ff in $> there holds ( 6 ^ ) i£ A1 B̂  d , Aá Bž Ĉ  are t r iangles such that Ai,A1£ k,b,ll3„3* then ^ / 1 V . Î gmjaaJL* ket ® translat ion affine plane íP sat isfy (l)« Then (4) holds in TL* iff, in <P , i t holds (7 ) I f Af B̂  Cj J>1} ^2&2C2P2 are parallelograms such that ^Af\At€ 02*> &1'C1> Bž 6 0 W C N t h e i d e a l p o i n t o f t h e l ine 3„ 3* ) \ ^VJ/(^\IIQ3„s A^fA^\ # 03 + then B4 e ON . Proposition 2» Let J& be a t ranslat ion affine plane satisfying ( l ) and ( 6 ^ ) . Then (6^. ) i s valid for a l l f r a -mes fm 03^ 3 3^ with J £ 0 J * . LfFM 4- ket ^ be an affine plane with a fixed f rá-me íT* m 03^ 3* J * , Then the Mright inverse property* ' í 8 j » J Ca, Ar')fr «• <*, for a i l a s T^ } Jlre \^<0\ . i s sa t i s f ied in TfM iff, in í5 f there holds (9p>m ) If A B> (̂  J3* } Ag b>Ř t^ fy are paraaielograms such tha t A1 B, / C, D, i AĚ\/ CĚ\t0 3M\ A, P< / &, $ / Mz *t / CÉPŘ / 03* 5 Vi^<?*H VA ti^Oj^) % € ̂ , A,C,-- M2CŽ « 03* then ^ € 01>f - 464 Proposition 3. Let (P be an affine plane satisfying (1) and (9*,*). Then (9p ) holds for all frames P m e OJ JQut \ e 0 ^ iff the "general right inverse pro-perty" is valid in Tflh t (10^# ) (Ca,c )Um4Jr) • C - a (><• ) for all o,,Jtr€TT* and c 6 T̂ ,* \ *f 0 } • Rem ark» If "Tl* possesses associative multiplication then ClO-,* ) is flilfilled. Moreover, if Tf* is an alter-native field, (10-* ) is satisfied. Further, the associati-vity of multiplication in 7L+ is equivalent to (11 j * ) (a,c)(c''14r) s a,4r for al l O/, & e T^m $ c e Tf*\ iOl -frplTM A*- ^et ^ he an affine plane with a fixed frame 5"** 0 0^ J* J * . Then, in "p* , there holds (8^. ) a/(a.4r) m Jtr for a l l a « ^ \ { f l i j > 6 \ + i f f P̂ sat i s f ies (9^. ) If A B C J> ; ^ B C J? are parallelograms such that AibJl^/tA^fC^/OJ*; A, Pi /RfCi lA^/B^CJ 10% \ fi^C,- 4D 2 i A, « fl-W°%i c** of* th*n ^i^°\' Proposition 3# . Let iP be an affine plane with a f i -xed frame f*m 0J^ 3? J* and let ( 8 ^ ) , (8 ' ) be sa t i s -fied • Then (8^, ) holds for a l l frames fm 0J# J JL with J%€ 0 J* . Definition 1. Let ^ be a translation affine plane satisfying ( l ) . Let TJ^ satisfy the condition 1 + i * 0 . If AtB,C are pairwise distinct points on the coordinate axis 0 J„ such that C + MA0 (.the "middle point" - 465 -of A,B ) the t r i p l e (AtB,C) w i l l be ca l led an admissible tr io le> To each admissible t r i p l e (A,B,C) we asšociate the point *""U&C i» "the f ollowing manner: Write A ar-s ( a , 0 ) , B « ( 4 r , 0 ) , Cs(c90) with respect to T^* and construct the points Sbn DmV * 8-f ^ C ^ O ^ V ^ C , ( X the ideá l point of 0D£ , $ ** (<Z, 4 ) ) , further the point P^ such that B«j « M^ p and f i n a l l y the point Hf*c * $T>4 rs OJ* . ProoosJt ion 4f From the assumptions of Def in i t ion 1 i t fo l lows H j 0 C m H**t for a l l framess T'• 0 Dx 3 3^ with X £ 0 J and for a l l admissible t r i p l e s (A f B f C) . Lemma 6» Let P be a t rans la t ion aff ine plane s a t i s -fying ( 1 ) , C 6 ^ ) t ( 9 ^ * ) and 1 + U í l in Tp* • Then for A m M , 0 > , h -(-4,0), C*(c,0) ± ( 0 , 0 ) i t f o l -i o * * H?* * < V ' , 0> . ABC 7 Def ini t ion £« Let íP be a trans lat ion affine plane sa-t i s f y i n g the assumptions of Lemma 6. By a von Staudt pro.jec-tivit.Y on Q3X we sha l l mean a b i j ec t ion 6" of 03# onto i t s e l f preserving at both s ides a l l admissible tr iple s and a l l points ^ABC Cwhere (A,B,C) runs over a l l admissible t r i p l e s ) . Proposit ion 5. Let fi be a trans lat ion aff ine plane dat i s fy ing the assumption of Lemma 6. If ff i s a von Staudt p r o j e c t i v i t y of 03# with f ixed points 0} 3# then the mapping % : T^* —y ~Tp>#> def ined by A* * » C a # , 0) for a l l A « (CL, 0 ) 6 0 3^ s a t i s f i e s the conditions - 466 -li* ň* & -— (i^ ) (a, +&) '« a, ů + > # for a l l cu, A* e Tr+ y ( i i « ; ) f a - 4 ) * - - ( O , * ) - ' fo r a l l a € " £ » N < 0 J . Conversely, i f jD ? "T^* —• "T^* i s a b i j e c t i o n with f ixed elements 0 ,1 and if ( i ^ ) , ( i i ^ ) are f u l f i l l e d then the mapping jD# ; 0 J* —+ 0 J * defined by A • (af,0) for a l l A » Ca,, 0) 6 0 J * i s a von Staudt p r o j e c t i v i t y of 0 J * • x) To be published i n Czech.Math.Journal . H e f e r e n c e s t i ] R.H. BRUGK: fíecent advanees in the foundafcions of E u c l i -dean geometry, Amer.Math.Monthly 62(1955), No.7 ,Par t 11 ,1 -17 . (Received May 29,1967) ~ 467 -

One generalization of the fourth harmonic point (Preliminary communication)

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One generalization of the fourth harmonic point (Preliminary communication)

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Institute of Mathematics AS CR, v. v. i.