Question

b) What is a plane dual Geometry? Prove in IG the statement dual to the Incidence Axiom 3.

what is a plane dual Geometry? Prove in IG the statement dual to Incidence Axiom 3.

Answer 1

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ANSwhal is pane doel qsemeta A shiking-tofu e ch bryntice blames the Summetry o the roles plxyed by points in the defrniFions and theoems, plane dual geome s the fomalization his conceb There are tu abproaches to the Subjotof da one thrugh Languape Cprneiple & dual) and he other move tuncboal abppach thraugh special mabpings 9 the fanctional abprach there 1s a mab between late geometry tha大ǐ, called a-duality Brook th Bual to the Pziom 3 of ual Paiom 3 The exist af leark Hour lines, no Proot Trree of whivh are conturrent." Let A, B, andD be four distinct poits, no Hree coLlimear, the existence oHhese pointssgiven by ciom 3. Thusby Aciom 1( Bny two dishinct poimts ae inddentuith exactly oneli刀e) , and simce no three d the ants ave co llinear 3C, BD, and cD. Consicer the four lins A8,c, cD, anl DA We assert thaol no hree oh these lires are comuzent Suppose not, then threeo the rres wulol be concuvent , thoe ave shx dishint lines PB,AC, AD, AB, BC, and CD ape conunent. By the Dual Aruiom 1 Bis the only pointo interseronf 9B and Bc. Hente Bis the point ocoueney for tre thote Lines PB,8c an armumphon het B,c,and D ae noncallimeas. The oher cases foUuw om a similar exiot at leaul our Lines, no thees af ehich ave cemarert d CD. Thus B is on CD, which contradicts the mitia men