\documentclass[% version=last,% %a6paper,% 12pt,% %landscape,% %titlepage,% %headings=small,% %twoside=false,% parskip=half,% %draft=true,% %DIV=12,% %headinclude=false,% %pagesize ] {scrartcl} \usepackage{pstricks} \usepackage{pst-eucl,pst-plot} \usepackage{mxedruli} % * "latex " does not give the Georgian letters; % * "pdflatex -shell-escape " does, % but needs the pdf option with pstricks, % and even then rotates the figures 90 degrees clockwise; % * "xelatex "" works, though is very slow % (takes ~ 3'30", at home, 40" at the office)) % The following two packages did not help: %\usepackage{graphicx} %\usepackage{pdflscape} \usepackage[landscape,a6paper,margin=1cm]{geometry} \usepackage{relsize} \pagestyle{empty} %\setlength{\footskip}{1\baselineskip} \usepackage{moredefs,lips} \usepackage{graphicx} \usepackage{multicol} \setlength{\columnseprule}{0.4pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{hfoldsty,verbatim} \newenvironment{optional}{}{} %\let\optional\comment %\let\endoptional\endcomment \usepackage[neverdecrease]{paralist} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsmath,amsthm,amssymb,url} %\allowdisplaybreaks \newcommand{\lto}{\Rightarrow} \newcommand{\liff}{\Leftrightarrow} \newcommand{\rat}[2]{#1\mathbin:#2} \newcommand{\prop}{\mathrel{:\,:}} \newcommand{\Apex}{A} % Apex of axial triangle \newcommand{\Beat}{B} % Base endpoint of axial triangle \newcommand{\Com}{C} % Complementary endpoint of base of axial triangle \newcommand{\mbat}{m} % midpoint of base of axial triangle (not shown) \newcommand{\Dp}{D} % Distant [end] point of chord \newcommand{\Dpo}{D'} % Distant point (opposite) of chord \newcommand{\Edp}{E} % Extended diameter point \newcommand{\Endp}{E^*} % Extended new diameter point \newcommand{\Fp}{F} % Four-way crossing \newcommand{\Jp}{J} % [pro-] Jected point in base \newcommand{\Jop}{J'} % [pro-] Jected opposite point \newcommand{\Koc}{O} % Kentrum of conic \newcommand{\Lk}{L}% L point next to K \newcommand{\Lok}{L'}% L point (opposite) next to K \newcommand{\Lkn}{L^*}% new L point next to K \newcommand{\Mp}{M} % Midpoint of chord \newcommand{\Mpn}{M^*}% Midpoint of new chord \newcommand{\Nmp}{N} % New [projected] midpoint \newcommand{\Vern}{\Dp}% Vertex new \newcommand{\Pnt}{P} % Point on curve \newcommand{\Pnto}{P'}% Point on curve (opposite) \newcommand{\Qxr}{Q} % Q in line with X and R new axial triangle base \newcommand{\Rxq}{R} % R in line with X and Q \newcommand{\Ver}{V} % vertex of section \newcommand{\Vasn}{V^*}% Vertex as such (new) \newcommand{\Wv}{W} % Way-out (opposite) vertex \newcommand{\Wnv}{W^*} % Way-out (opposite) new vertex \newcommand{\Xp}{X} % X point (foot of ordinate} \newcommand{\Xpn}{X^*}% X point (new foot of ordinate) \newcommand{\Yp}{Y} % Y point (lined up with P and X*) \newcommand{\Ypn}{Y^*}% Y point (lined up with P and X) \newcommand{\hide}{ee} % hidden variables for constructions \newcommand{\hida}{a} \newcommand{\hidb}{b} \newcommand{\hidc}{c} \newcommand{\hidd}{d} % The following are for Desargues's Theorem \newcommand{\TA}{A} \newcommand{\TB}{B} \newcommand{\TC}{C} \newcommand{\TD}{D} \newcommand{\TE}{E} \newcommand{\TF}{F} \renewcommand{\vec}[2]{\overrightarrow{#1#2}} \renewcommand{\theta}{\vartheta} \renewcommand{\theequation}{\fnsymbol{equation}} \usepackage{bm} \renewcommand{\leq}{\leqslant} \renewcommand{\geq}{\geqslant} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \raggedright \begin{document} %\titlehead{\centering} \title{Affine Planes with Polygons} \author{David Pierce\\ Mimar Sinan G\"uzel Sanatlar \"Universitesi\\ % Istanbul\\ \url{mat.msgsu.edu.tr/~dpierce}, \url{polytropy.com}} \date{Batumi ({\mxedr batumi}), September, 2019} \publishers{\smaller{\mxedr sakartvelos matema.ti.kosta .kav+siris}\\ X {\mxedr qovel.cliuri saerta+soriso .konperencia}\\ X Annual International Conference\\ of the Georgian Mathematical Union} {\relscale{0.2}\maketitle} %\maketitle \newpage \thispagestyle{empty} \mbox{}\vfill \begin{center} \psset{dotsize=8pt,PointName=none} \newcommand{\myxrad}{6}\newcommand{\myyrad}{4.5} \begin{pspicture}(-5,-3)(5.5,3.5) \parametricplot[linewidth=2.4pt]0{360} {t cos \myxrad\space mul t sin \myyrad\space mul} \pstGeonode[PosAngle={180,-12,78,35}] % PointSymbol={default,default,default,default,o}] (0,0){\Koc} (! -12 cos \myxrad\space mul -12 sin \myyrad\space mul){\Ver} (! 90 -12 add cos \myxrad\space mul 90 -12 add sin \myyrad\space mul) {\Lk} (! 35 cos \myxrad\space mul 35 sin \myyrad\space mul){\Vasn} % (! -85 cos \myxrad\space mul -85 sin \myyrad\space mul){\Pnt} % \ncline{\Koc}{\Ver}\ncline{\Koc}{\Vasn} \uput[r](\Koc){${\Koc}$} \uput[-10](\Ver){${\Ver}$} \uput[80](\Lk){${\Lk}$} \uput[35](\Vasn){${\Vasn}$} \rput(0,2.3){\begin{minipage}{\textwidth}\centering Let \begin{equation*} {\Vasn}={\Koc}+a\cdot\vec{\Koc}{\Ver}+b\cdot\vec{\Koc}{\Lk} \end{equation*} arbitrarily on the locus of ${\Pnt}$, where \begin{gather*} {\Pnt}={\Koc}+x\cdot\vec{\Koc}{\Ver}+y\cdot\vec{\Koc}{\Lk},\qquad x^2\pm y^2=1. \end{gather*} \end{minipage}} \rput(0,-2.4){\begin{minipage}{\textwidth}\centering \textbf{Apollonius's Theorem.} The affinity fixing ${\Koc}$ and\\ interchanging ${\Ver}$ and ${\Vasn}$ fixes the locus.\\ \mbox{}\\ \textbf{Modern proof.} The affinity corresponds to \begin{equation*} \begin{pmatrix} x\\y \end{pmatrix}\mapsto \begin{pmatrix} a&\pm b\\b&-a \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix}. \end{equation*} \end{minipage}} \end{pspicture} \end{center} \vfill\mbox{} \newpage \textbf{Apollonius's Proof.} \hfill The locus is given by \fbox{${\Xp}{\Pnt}{\Yp} ={\Ver}{\Xp}{\Ypn}{\Endp}$,} \vfill\hfill because this is true when ${\Pnt}$ is ${\Vasn}$, and \vfill \hfill ${\Xp}{\Pnt}{\Yp} \propto{\Xp}{\Pnt}^2 \propto{\Xp}{\Ver}\cdot{\Xp}{\Wv} \propto{\Ver}{\Xp}{\Ypn}{\Endp}$. \vfill \newcommand{\myxrad}{3.5}\newcommand{\myyrad}{2.3} %\psset{unit=9mm} \begin{pspicture}(-4,-1.1)(3,2.7) %\psgrid \psset{PointSymbol=none} \parametricplot[linewidth=2.4pt]0{360}{t cos \myxrad\space mul t sin \myyrad\space mul} \pstGeonode[PosAngle={105,-60,78,75,0,0,-85}, PointName={default,default,default,default,none,none,default}] (0,0){\Koc} (! -12 cos \myxrad\space mul -12 sin \myyrad\space mul){\Ver} (! 90 -12 add cos \myxrad\space mul 90 -12 add sin \myyrad\space mul){\Lk} (! 35 cos \myxrad\space mul 35 sin \myyrad\space mul){\Vasn} (! 0 47 cos)x (0,1)t (! -85 cos \myxrad\space mul -85 sin \myyrad\space mul){\Pnt} \pstTranslation[PointName=none] t{\Ver}x \pstInterLL[PosAngle=-105]{x'}x{\Koc}{\Ver}{\Mp} \pstTranslation[PointName=none] x{\Ver}t \pstInterLL[PosAngle=-60]{t'}t{\Koc}{\Ver}{\Edp} \pstTranslation[PointName=none]{\Mp}{\Vasn}{\Ver,\Pnt}[u,v] \pstInterLL[PosAngle=90]{\Ver}u{\Koc}{\Vasn}{\Endp} \pstInterLL[PosAngle=-60]{\Pnt}v{\Koc}{\Ver}{\Xp} \pstInterLL[PosAngle=90]{\Pnt}{\Xp}{\Koc}{\Vasn}{\Ypn} \pstTranslation[PointName=none]{\Edp}{\Vasn}{\Pnt}[w] \pstInterLL[PosAngle=75]{\Pnt}w{\Koc}{\Ver}{\Yp} \pstInterLL[PosAngle=-105]{\Pnt}{\Yp}{\Koc}{\Vasn}{\Xpn} \pstTranslation[PosAngle=150]{\Ver}{\Koc}{\Koc}[\Wv] \pstTranslation[PosAngle=-150,PointName=none]{\Vasn}{\Koc}{\Koc}[\Wnv] % \pstInterLL[PosAngle=10,PointName=none]{\Vasn}{\Edp}{\Ver}{\Endp}{\Fp} \ncline[linewidth=2.4pt]{\Wv}{\Edp} \ncline{\Mp}{\Vasn} \ncline{\Ver}{\Endp} \ncline{\Pnt}{\Ypn} \ncline{\Koc}{\Lk} \psset{linestyle=dashed} \ncline{\Vasn}{\Edp} \ncline{\Pnt}{\Yp} \psset{linestyle=dotted} \ncline{\Wnv}{\Endp} \end{pspicture} \vfill\hfill Adding ${\Xp}{\Yp}{\Xpn}{\Ypn}$ yields \vfill\hfill ${\Ypn}{\Pnt}{\Xpn} ={\Ver}{\Yp}{\Xpn}{\Endp}$, \vfill\hfill then \fbox{${\Ypn}{\Pnt}{\Xpn} ={\Edp}{\Yp}{\Xpn}{\Vasn}$.} \newpage \begin{multicols}{2} The foregoing happens in an \textbf{affine plane,} satisfying \begin{asparaenum}[1)] \item two points determine a line; \item through a point not on a line, a single parallel passes; \item there is a proper triangle. \end{asparaenum} \begin{pspicture}(3,3.6) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={-90,-90,-45,90}] (2,0.6){\TE}(0.2,1.2){\TD}(2.8,1.5){\TF}(2.6,3){\TC} % \psgrid \pstHomO[HomCoef=3,PointName=none]{\TF}{\TC}[O] \pstTranslation[PointName=none]{\TF}{\TD}{\TC}[x] \pstInterLL[PosAngle=135]{\TC}x{\TD}O{\TA} \pstTranslation[PointName=none]{\TF}{\TE}{\TC}[x] \pstInterLL[PosAngle=-135]{\TC}x{\TE}O{\TB} % \psset{linestyle=dashed} \ncline{\TB}{\TE}\ncline{\TA}{\TD}\ncline{\TC}{\TF} \psset{linewidth=2.4pt} % \psset{linestyle=solid} \ncline {\TD}{\TE}\ncline {\TD}{\TF} \ncline {\TB}{\TA}\ncline {\TA}{\TC} \psset{linestyle=dashed} \ncline{\TF}{\TE} \ncline{\TC}{\TB} \end{pspicture} \hfill \begin{pspicture}(2.6,3.6) \psset{PointSymbol=none} % \psgrid \pstGeonode[PosAngle={-90,-90,90,90}] (0,0.6)D(2.6,0.6){\TE}(1.2,3){\TB}(2,3)A \pstHomO[HomCoef=0.45,PosAngle=150]{\TB}D[{\TC}] \pstTranslation[PointSymbol=none,PointName=none] {\TC}{\TE}{\TB}[x] \pstInterLL {\TB}xA{\TE}{\TF} \ncline D{\TB}\ncline A{\TE} \ncline{\TB}{\TD}\ncline{\TA}{\TE} \psset{linewidth=2.4pt} \ncline {\TD}{\TE}\ncline {\TD}{\TF} \ncline {\TB}{\TA}\ncline {\TA}{\TC} \psset{linestyle=dashed} \ncline{\TF}{\TB} \ncline{\TC}{\TE} \end{pspicture} The plane is $K^2$ for some field $K$, if also, assuming \begin{equation*} {\TA}{\TB}\parallel{\TD}{\TE}\And{\TA}{\TC}\parallel{\TD}{\TF}, \end{equation*} %we require \begin{asparaenum}[1)]\setcounter{enumi}{3} \item \textbf{Desargues's Theorem:} \begin{equation*} {\TB}{\TC}\parallel {\TE}{\TF}, \end{equation*} if ${\TA}{\TD}$, ${\TB}{\TE}$, and ${\TC}{\TF}$ either \begin{compactenum} \item are mutually parallel or \item have a common point; \end{compactenum} \item \textbf{Pappus's Theorem:} \begin{equation*} {\TB}{\TF}\parallel {\TC}{\TE}, \end{equation*} if \parbox{5cm}{% \begin{compactitem} \item ${\TD}$ lies on ${\TB}{\TC}$ and \item ${\TA}$ lies on ${\TE}{\TF}$. \end{compactitem}} \end{asparaenum} % Now we need a theory of \emph{areas.} \end{multicols} \newpage \begin{multicols}{2} Of Desargues, case (a), \textbf{``Prism,''} lets us define, for non-collinear directed segments, % $AD$ and $BE$, \begin{multline*} \vec AD=\vec BE \iff\\ ABED \text{ is a parallelogram;} \end{multline*} \begin{center} \begin{pspicture}(5,3.6) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={-30,180,0,0}] (3,0.3){\TD}(0.6,1){\TE}(4.4,1.7){\TF}(4.4,3.4){\TC} % \psgrid \pstTranslation[PosAngle={180,-30}] {\TF}{\TC}{{\TE},{\TD}}[{\TB},{\TA}] \ncline {\TE}{\TD}\ncline[linestyle=dotted] {\TE}{\TF}\ncline {\TF}{\TD} \ncline {\TA}{\TB}\ncline[linestyle=dotted] {\TB}{\TC}\ncline {\TC}{\TA} {\psset{linewidth=2.4pt} \ncline {\TA}{\TD}\ncline {\TB}{\TE}} \psset{linestyle=dashed} \ncline {\TC}{\TF} \end{pspicture} \end{center} case (b), \textbf{``Pyramid,''} for non-parallel pairs of parallel vectors, \begin{multline*} \rat{\vec OA}{\vec OD} \prop\rat{\vec OB}{\vec OE}\\ \iff AB\parallel DE. \end{multline*} \begin{center} \begin{pspicture}(5,3.6) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={-30,180,0,90}] (3,0.3){\TD}(0.6,1){\TE}(4.4,1.7){\TF}(2.2,3)O \pstHomO[HomCoef=0.4,PosAngle={-135,165,45}]O{{\TD},{\TE},{\TF}}[{\TA},{\TB},{\TC}] \ncline {\TE}{\TD}\ncline[linestyle=dotted] {\TE}{\TF}\ncline {\TF}{\TD} \ncline {\TA}{\TB}\ncline[linestyle=dotted] {\TB}{\TC}\ncline {\TC}{\TA} {\psset{linewidth=2.4pt} \ncline O{\TD}\ncline O{\TE}} \psset{linestyle=dashed} \ncline O{\TF} \end{pspicture} \end{center} On the plane, ratios of vectors act as a field or skew-field. With Pappus, it is a field. \end{multicols} \newpage \begin{multicols}{2} For Pappus and Desargues to be \emph{Theorems,} I propose axioms: \begin{asparadesc} \item[Addition.]\label{ax:group} The polygons compose an abelian group where \begin{gather*} -ABC\cdots= \cdots CBA,\\ AAB=0,\qquad A*{}={}* A,\\ A* B+B\mathrel{\dag} A= A* B\mathrel{\dag}, \end{gather*} $*$ and $\dag$ being strings of vertices. \item[Linearity.]\label{ax:col} % $ABC=0$ means $A$, $B$, and $C$ are collinear: \begin{multline*} ABC\neq0\land BCD=0\\ {}\land C\neq D\lto ACD\neq0. \end{multline*} \item[Parallels\lips]\label{ax:Playfair} % Playfair's Axiom. \item[Translation.]\label{ax:2-par} $ABED$ and $BCFE$ being parallelograms, \begin{equation*} ABC=DEF. \end{equation*} \begin{pspicture}(4.8,3.2) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={180,90,-90}] (0.6,1.1)G(1,2.6)A(1.5,0.6)C \pstGeonode[PointName=none](3,1.6)O \pstSymO[PosAngle={-90,0,90}] O{A,G,C}[F,E,D] \pstTranslation[PosAngle=-45] DAE[B] \ncline AD\ncline CF\ncline BE \psset{linestyle=dotted} \psset{linestyle=dashed} % \ncline FG\ncline BG % \ncline DA\ncline DE \psset{linestyle=solid,linewidth=2.4pt} \ncline AC\ncline CB\ncline BA \ncline DF\ncline FE\ncline ED \ncline AG\ncline CG \end{pspicture} \item[Bisection.]\label{ax:1-par} $ABCG$ being a parallelogram, \begin{equation*} CGA=ABC. \end{equation*} \item[Halving.]\label{ax:not-2} All nonzero polygons have the same order, not $2$. \end{asparadesc} \end{multicols} \newpage \begin{multicols}{2} Hence \textbf{Euclid \textsc i.37, 39:} \begin{pspicture}(4.8,4) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={-90,-90,90,90}](1.1,0.6)C(3.5,0.6)D (0.2,3.4)A \pstTranslation[PosAngle=90]CDA[B] \pstHomO[HomCoef=1.2,PosAngle=90]AB[E] \pstTranslation[PosAngle=90]CDE[F] \pstInterLL[PosAngle=-95] CEDBG {\psset{linewidth=2.4pt} \ncline AD\ncline DC\ncline CA \ncline CF\ncline FD\ncline DC} \ncline AF % \psset{linestyle=dashed} \ncline BD\ncline CE \pstHomO[HomCoef=0.8,PosAngle=-30]DF[H] \psset{linestyle=dotted} \ncline AH\ncline HC % \pstHomO[HomCoef=1.8,PosAngle=-90]CD[K] % \ncline[linestyle=dashed]DK \end{pspicture} $\bullet$ Assuming $AF\parallel CD$, by \textbf{Parallels} we may let \begin{align*} AC&\parallel BD, &CE&\parallel DF; \end{align*} by \textbf{Translation,} \begin{equation*} ACE=BDF; \end{equation*} by \textbf{Addition,} \begin{comment} \begin{equation*} ACDB=ECDF, \end{equation*} since \end{comment} \begin{multline*} ACDB =ACGB+CDG={}\\ ACE-BGE+CDG={}\\ BDF-BGE+CDG =ECDF; \end{multline*} by \textbf{Bisection,} \begin{gather*} ACDB=2ACD,\\ ECDF=2FCD; \end{gather*} by \textbf{Halving,} \begin{equation*}%\label{eqn:ACD} ACD=FCD. \end{equation*} $\bullet$ If $AF\nparallel CD$, let $AH\parallel CD$. \begin{align*} ACD&=HCD,\\ FCD&=HCD+FCH, \end{align*} so $ACD\neq FCD$ %\eqref{eqn:ACD} fails by \textbf{Linearity.} \end{multicols} \newpage \begin{minipage}{9cm} We can now prove: \begin{compactitem} \item \textbf{Prism,} by Translation, \textsc i.39, and Parallels. \item \textbf{Pappus.} From ${\TA}{\TB}\parallel{\TD}{\TE}$ and ${\TA}{\TC}\parallel{\TD}{\TF}$, we obtain ${\TB}{\TF}\parallel{\TC}{\TE}$, since, by \textsc i.37, \begin{equation*} {\TB}{\TF}{\TC} ={\TB}{\TF}{\TA}{\TD} ={\TB}{\TF}{\TE}. \end{equation*} \end{compactitem} \end{minipage} \hfill \begin{minipage}{3cm} \begin{pspicture}(3,3) \psset{PointSymbol=none} % \psgrid \pstGeonode[PosAngle={-150,-30,180,0}] (0,0){\TC}(2.6,0){\TE}(1,2.8){\TB}(2,2.8){\TF} \pstHomO[HomCoef=0.45,PosAngle=150]{\TB}{\TC}[{\TD}] \pstTranslation[PointSymbol=none,PointName=none] {\TD}{\TE}{\TB}[x] \pstInterLL {\TB}x{\TF}{\TE}{\TA} \ncline {\TC}{\TB}\ncline {\TF}{\TE} \ncline{\TB}{\TC}\ncline{\TF}{\TE} %\psset{linestyle=dotted} \ncline{\TF}{\TC}\ncline{\TB}{\TE}\ncline{\TD}{\TA} \psset{linestyle=solid} \psset{linewidth=2.4pt} \ncline{\TA}{\TB} \ncline {\TC}{\TA} \ncline{\TD}{\TE} \ncline {\TF}{\TD} \psset{linestyle=dashed} \ncline {\TC}{\TE} \ncline {\TB}{\TF} \end{pspicture} \end{minipage} \begin{minipage}{4.5cm} \begin{pspicture}(4.5,5) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={0,180,180}] (3.9,4.8){\TE}(0.6,4)O(0.6,0.3){\TF} \pstHomO[HomCoef=0.45,PosAngle=180]O{\TF}[{\TC}] \pstHomO[HomCoef=0.35,PosAngle=135]O{\TE}[{\TB}] \pstTranslation[PosAngle=0] O{\TF}{\TE}[{\TD}] \pstTranslation[PosAngle=-45] O{\TF}{\TB}[M] \pstTranslation[PosAngle=0] O{\TE}{\TC}[N] \pstInterLL[PosAngle=135] {\TB}M{\TC}N{\TA} \psset{PointName=none} \pstTranslation[PosAngle=30]{\TB}{\TC}{\TE}[P] \pstTranslation[PosAngle=75] {\TC}{\TB}{\TF}[Q] \pspolygon[fillstyle=solid,fillcolor=lightgray] ({\TB})({\TE})(N)({\TA}) \pspolygon[fillstyle=solid,fillcolor=lightgray] ({\TC})({\TF})(M)({\TA}) {\psset{linewidth=2.4pt} \pspolygon(O)({\TE})({\TD})({\TF}) \ncline {\TB}M\ncline {\TC}N \psset{linestyle=dashed} \ncline {\TB}{\TC}\ncline {\TE}P\ncline Q{\TF}} % \ncline O{\TA} % \ncline PQ % \psset{linestyle=dashed} \ncline {\TA}{\TD} \pstMiddleAB {\TB}N{Ma} \rput*(Ma){$\beta$} \pstMiddleAB {\TC}M{Mb} \rput*(Mb){$\gamma$} \end{pspicture} \end{minipage} \hfill \begin{minipage}{7.5cm} $\bullet$ \textbf{Pyramid, special case.} Assume \begin{compactitem}[--] \item $BE$ and $CF$ meet at $O$, \item $AB\parallel DE$ and $AC\parallel DF$, \item ${\TA}{\TB}\parallel O{\TC}$ and ${\TA}{\TC}\parallel O{\TB}$. \end{compactitem} Then \begin{align*} {\TA}{\TD} \text{ contains }O &\iff\beta=\gamma\\ &\iff BC\parallel DF. \end{align*} \begin{comment} \begin{gather}\label{eqn:cont} {\TA}{\TD} \text{ contains }O,\\\label{eqn:par} AB\parallel DE\And AC\parallel DF \end{gather} imply \begin{equation}\label{eqn:BC||DF} BC\parallel DF. \end{equation} If \eqref{eqn:par}, \textbf{in case} ${\TA}{\TB}\parallel O{\TC}$ and ${\TA}{\TC}\parallel O{\TB}$, \begin{equation*} \eqref{eqn:cont} \iff \beta=\gamma \iff \eqref{eqn:BC||DF}. \end{equation*} \end{comment} \end{minipage} \newpage \begin{minipage}{7cm} \textbf{Pyramid, less special case.} If \begin{compactitem}[--] \item ${\TA}{\TD}$, ${\TB}{\TE}$, and ${\TC}{\TF}$ meet at $O$, \item $AB\parallel DE$ and $AC\parallel DF$, \item $AB\parallel OC$,\hfill then $BC\parallel DF$. \end{compactitem} \vfill \textbf{General case.} Assuming $AC'\parallel DF'$, \begin{equation*} ABC\sim DEF\implies ABC'\sim DEF', \end{equation*} provided $CC'\parallel OA$; we can remove this; putting $C'$ on $O{\TC}$ is enough. \begin{pspicture}(7,2.5) \psset{PointSymbol=none} % \psgrid \pstGeonode[PosAngle={-90,-90,180,0}] (0.2,0.3)O(6.8,0.3){\TE}(2,2.1){F'}(5.8,2.1){\TF} \pstHomO[HomCoef=0.7,PosAngle=-90]O{\TE}[{\TD}] % \pstTranslation[PosAngle=-90]{F'}O{\TF}[{\TD}] \pstHomO[HomCoef=0.4,PosAngle=135]O{\TF}[{\TC}] \pstTranslation[PointName=none] {\TF}{F'}{\TC}[x] \pstInterLL[PosAngle=180] {\TC}xO{F'}{C'} \pstTranslation[PointName=none] {\TF}{\TE}{\TC}[x] \pstInterLL[PosAngle=-90] {\TC}xO{\TE}{\TB} \pstTranslation[PointName=none]{\TF}{\TD}{\TC}[x] \pstInterLL[PosAngle=-90] {\TC}xO{\TE}{\TA} %%%%%%%%%%%%%%%%%%%%%%%% \pspolygon[linestyle=none,fillstyle=solid,fillcolor=lightgray] ({\TD})({\TF})({\TE}) \pspolygon[linestyle=none,fillstyle=solid,fillcolor=lightgray] ({\TA})({\TC})({\TB}) \pspolygon[linestyle=none,fillstyle=hlines] ({\TD})(F')({\TE}) \pspolygon[linestyle=none,fillstyle=hlines] ({\TA})(C')({\TB}) \ncline O{\TE} \ncline O{\TF}\ncline O{F'} \ncline{\TF}{F'} \ncline{\TC}{C'} \ncline{\TA}{C'}\ncline{C'}{\TB} \ncline{\TD}{F'}\ncline{F'}{\TE} \ncline{\TA}{\TC}\ncline{\TC}{\TB} \ncline{\TD}{\TF}\ncline{\TF}{\TE} \end{pspicture} \end{minipage} \hfill \begin{minipage}{4cm} \begin{pspicture}(0.7,-1.5)(4.7,7) % \psgrid \psset{PointSymbol=none} \pstGeonode[PosAngle={90,90,-90}] (4.5,6.5){\TE}(0.2,5.8)O(0.2,1.5){\TF} \pstHomO[HomCoef=0.35,PosAngle=90]O{\TE}[{\TB}] \pstTranslation[PointName=none]{\TE}{\TF}{\TB}[x] \pstInterLL[PosAngle=180] {\TB}xO{\TF}{\TC} \pstTranslation[PointName=none] O{\TF}{\TE}[P] \pstHomO[HomCoef=0.48]{\TE}P[{\TD}] \pstTranslation[PointName=none] O{\TD}{\TF}[Q] \pstTranslation[PointName=none] O{\TF}{\TB}[R] \pstInterLL[PointName=none] {\TB}RO{\TD}{\TA} % \pstInterLL[PosAngle=115] {\TE}{\TF}O{\TD}G % \pstInterLL[PosAngle=-170] {\TB}{\TC}O{\TD}H {\psset{PointName=none} \pstInterLL {\TB}R{\TF}QL \pstTranslation {\TF}P{\TC}[M] \pstTranslation {\TF}Q{\TC}[N] \pstTranslation {\TE}{\TA}H[x] % \pstInterLL[PosAngle=90] HxO{\TE}K \pstInterLL{\TB}LCNS \pstInterLL{\TC}M{\TB}LT} \pspolygon[linestyle=none,fillstyle=solid,fillcolor=lightgray]({\TC})({\TF})(L)(S) \pspolygon[linestyle=none,fillstyle=hlines]({\TC})({\TF})(R)(T) \pspolygon[linestyle=none,fillstyle=solid,fillcolor=lightgray]({\TA})(S)(N)({\TD}) \pspolygon[linestyle=none,fillstyle=hlines]({\TB})(T)(M)(E) \uput*[30]({\TA}){${\TA}$} \ncline {\TF}P\ncline {\TD}Q\ncline Q{\TF}\ncline {\TA}L \ncline {\TC}M\ncline {\TC}N % {\psset{linestyle=dotted} % \ncline HK\ncline KG\ncline {\TE}{\TA}} \psset{linewidth=2.4pt} \ncline O{\TE}\ncline O{\TF}\ncline O{\TD} \ncline {\TE}{\TD}\ncline {\TB}{\TA} \ncline {\TF}{\TD}\ncline {\TC}{\TA} \psset{linestyle=dashed} \ncline {\TE}{\TF}\ncline {\TB}{\TC} \psset{linestyle=dotted} \end{pspicture} \end{minipage} \newpage \begin{minipage}[c][8cm]{7.5cm} Given \fbox{${\TA}{\TB}{\TC}\sim {\TD}{\TE}{\TF}$,} %for arbitrary $C'$ on ${\TC}{\TF}$ we let \begin{gather*} % {\TD}{F'}\parallel{\TA}{C'},\\ {\TB}G\parallel{\TA}{\TC},{\TD}{\TF},\\ HG,{\TD}{F'}\parallel{\TA}{C'}. \end{gather*} By Pappus, \begin{enumerate}[1)] \item from ${\TA}{\TC}HG{\TB}{C'}$, $H{\TC}\parallel {\TB}{C'}$; \item from ${\TB}{\TC}{\TD}{\TF}{\TE}G$, ${\TD}{\TC}\parallel{\TE}G$; \item from $H{\TC}{\TD}{F'}{\TE}G$, $H{\TC}\parallel {\TE}{F'}$. \end{enumerate} Thus \begin{equation*} {\TB}C'\parallel{\TE}F', \end{equation*} \fbox{${\TA}{\TB}{C'}\sim {\TD}{\TE}{F'}$;} also \fbox{${\TA}{\TD}{\TC}\sim {\TB}{\TE}G$.} \end{minipage} \hfill \begin{minipage}{4.5cm} \begin{pspicture}(0.5,0)(5,8.5) % \psgrid[subgriddiv=1] \psset{PointSymbol=none} \pstGeonode[PosAngle={180,0,180,0}] (0,8.3){\TA}(4.5,8){\TC}(1.5,0.2){\TE}(4.5,1.5){F'} \pstHomO[HomCoef=0.65,PosAngle=180]{\TA}{\TE}[{\TD}] \pstTranslation[PointName=none] {\TD}{\TC}{\TE}[x] \pstInterLL[PosAngle=0] {\TE}x{\TC}{F'}G \pstTranslation[PointName=none] {\TC}{\TA}G[x] \pstInterLL[PosAngle=180] Gx{\TA}{\TE}{\TB} \pstTranslation[PointName=none] {\TC}{\TA}{\TD}[x] \pstInterLL[PosAngle=0] {\TD}x{\TC}{F'}{\TF} \pstTranslation[PointName=none] {\TE}{F'}{\TB}[x] \pstInterLL[PosAngle=0] {\TB}x{\TC}{F'}{C'} \pstTranslation[PointName=none] {F'}{\TE}{\TC}[x] \pstInterLL[PosAngle=180] {\TC}x{\TA}{\TE}H %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \psset{fillstyle=solid,fillcolor=lightgray} \pspolygon (H)({\TC})({\TD})({F'})({\TE})(G) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \psset{hatchangle=41,fillcolor=gray} \pspolygon%[fillstyle=hlines] ({\TB})({\TC})({\TD})({\TF})({\TE})(G) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \psset{hatchangle=!0.3 4.5 atan neg} \pspolygon[fillstyle=hlines] (H)({\TC})({\TA})({C'})({\TB})(G) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \psset{linestyle=solid} \ncline{\TA}{\TE}\ncline {\TC}{F'} % \psset{linestyle=dashed} % \ncline {\TD}{\TC}\ncline {\TE}G % \psset{linestyle=dotted} % \ncline H{\TC}\ncline {\TB}{C'}\ncline {\TE}{F'} \psset{linewidth=2.4pt} \ncline HG\ncline {\TA}{C'}\ncline {\TD}{F'} \ncline{\TA}{\TC}\ncline{\TB}G\ncline{\TD}{\TF} \end{pspicture} \end{minipage} \end{document}