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0 0 }0 0 0 -1 -1 -1 3 6 0 0 0 0 0 0 }{PSTYLE "sub problem" 0 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "diagram" -1 295 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 " The Harmonic rays theore m." }}{EXCHG {PARA 289 "" 0 "" {TEXT 261 23 "Theorem (harmonic rays)" }{TEXT -1 70 ": Let ABCD be a quadrilateral. Let F be the point of in tersection of " }{XPPEDIT 18 0 "line AD" "*&%%lineG\"\"\"%#ADGF$" } {TEXT -1 6 " with " }{XPPEDIT 18 0 "line BC" "*&%%lineG\"\"\"%#BCGF$" }{TEXT -1 45 " and let G be the point of intersection of " } {XPPEDIT 18 0 "line AB" "*&%%lineG\"\"\"%#ABGF$" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "line CD" "*&%%lineG\"\"\"%#CDGF$" }{TEXT -1 75 " . One of both of the points F and G may be on the line at infinity. Let \+ " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 37 " be the line through A \+ parallel to " }{XPPEDIT 18 0 "line FG" "*&%%lineG\"\"\"%#FGGF$" } {TEXT -1 60 " and let M be the intersection of the diagonals of ABCD. \+ If " }{XPPEDIT 18 0 "line BF" "*&%%lineG\"\"\"%#BFGF$" }{TEXT -1 8 " \+ meets " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 12 " at P and " } {XPPEDIT 18 0 "line MF" "*&%%lineG\"\"\"%#MFGF$" }{TEXT -1 8 " meets \+ " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 53 " at Q then the line segm ents PQ and QA are congruent." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "A diagram " }}{PARA 0 "" 0 "" {TEXT -1 133 " (execute these to set up diagram unless the geometry package setup for the theorem on shadows \+ of squares has already been executed )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a1 =0: a2 =0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "b 1:=-6: b2:=-3/2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c1:=-4: c2:=-9: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d1:=3: d2:=-3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "geometry[point](A,[a1,a2]):\ngeometry[point](B, [b1,b2]):\ngeometry[point](C,[c1,c2]):\ngeometry[point](D,[d1,d2]):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "geometry[segment](AB,[A,B]):\ngeo metry[segment](AC,[A,C]):\ngeometry[segment](BC,[B,C]):\ngeometry[segm ent](CD,[C,D]):\ngeometry[segment](AD,[A,D]):\ngeometry[line](lineAD,[ A,D]):\ngeometry[segment](BD,[B,D]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "geometry[line](lineCD,[C,D]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "geometry[line](lineBC,[B,C]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "geometry[line](lineAD,[A,D]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " geometry[line](lineBD,[B,D]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "geometry[line](lineAC,[A,C]):\ngeometry[line](lineAB,[A,B]):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "geometry[intersection](F, lineBC,li neAD):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "geometry[intersection](M, lineBD,lineAC):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "geometry[inters ection](G, lineAB,lineCD):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "geome try[line](lineFM,[F,M]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "geometr y[line](lineGM,[G,M]):\ngeometry[line](vanishingline,[F,G]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "axis:='axis':geometry[ParallelLine](axis, A,vanishingline):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T:='T':geometr y[intersection](T, lineBD,axis):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Q:='Q': geometry[intersection](Q, lineFM,axis):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "R:='R':geometry[intersection](R, lineGM, axis):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "geometry[intersection](P, axis,line BC):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "harmonicdiagram:=\{axis( color=tan, thickness=2), vanishingline(color=tan, thickness=2), P,Q,A, lineAD(color=blue), AC(color=red), BD(color=red),F,G,lineAB(color=blu e),lineCD(color=blue),BC(color=blue,thickness=3),B,C,D,M,lineBC(color= blue),AD(color=blue,thickness=3),CD(color=blue,thickness=3),AB(color=b lue,thickness=3), lineFM(color=green)\}:" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 5 "PROOF" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "First we draw the diagram. When F or G is at inf inity then ABCD has at least one pair of parallel sides. 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With t =1 we have a point other than A on this line" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Tpt:=Apt+(Fpt-Gpt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$TptG7$,&*&,&*&%#c2G\"\"\"%#d1GF+!\"\"*&%#c1GF+%#d2GF +F+F+,&F*F+F0F-F-F+*(F*F+F,F+,(F)F-F0F-F.F+F-F-,$*(F*F+F0F+F3F-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Now we calculate P and Q, the poin ts of intersection of this parallel with the lines BC and MF" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Ppt:=intsect(Bpt,Cpt,Apt,Tpt );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$PptG7$*&,.*&%#d1G\"\"#%#c2GF* \"\"\"*&%#c1GF,%#d2GF*!\"\"*&F.F*F/F*F,**F)F,F+F,F.F,F/F,!\"#*(F/F,F+F ,F)F,F**&F+F*F)F,F0F,,4F1F,F-F3*(F+F,F.F,F/F,F,F2F3*$F/F*F,*&F+F,F/F,F 0F4F*F(F,F5F0F0,$**F+F,F/F,,&F+F,F/F0F,F6F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Qpt:=intsect(Mpt,Fpt,Apt,Tpt);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$QptG7$,$*(,.*&%#d1G\"\"#%#c2GF+\"\"\"*&%#c1GF-%#d 2GF+!\"\"*&F/F+F0F+F-**F*F-F,F-F/F-F0F-!\"#*(F0F-F,F-F*F-F+*&F,F+F*F-F 1F-,(*&F,F-F*F-F1F0F1*&F/F-F0F-F-F1,*F9F-F0F1F8F1F,F-F1#F-F+,$*,F,F-F0 F-,&F,F-F0F1F-F7F1F:F1#F1F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 299 "T he theorem says that the distance from P to Q is equal to the distance from Q to A. Since we have set A to be the origin this is equivalent \+ to saying that the distance from P to the origin is twice the distance from Q to the origin which is equivalent to \"ratsq\" the ratio of t he squares being 4. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rat sq:=(Ppt[1]^2+Ppt[2]^2)/(Qpt[1]^2+Qpt[2]^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ratsqG*&,&*&,.*&%#d1G\"\"#%#c2GF+\"\"\"*&%#c1GF-%#d2 GF+!\"\"*&F/F+F0F+F-**F*F-F,F-F/F-F0F-!\"#*(F0F-F,F-F*F-F+*&F,F+F*F-F1 F+,4F2F-F.F4*(F,F-F/F-F0F-F-F3F4*$F0F+F-*&F,F-F0F-F1F5F+F)F-F6F1F4F-** F,F+F0F+,&F,F-F0F1F+F7F4F-F-,&*(F(F+,(*&F,F-F*F-F1F0F1*&F/F-F0F-F-F4,* FAF-F0F1F@F1F,F-F4#F-\"\"%*,F,F+F0F+F " 0 "" {MPLTEXT 1 0 24 "simplify(expand(ratsq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "This completes the proof." }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "EXERCISES" }}{EXCHG {PARA 290 "" 0 "" {TEXT -1 235 "EXERCISE: Explain why statement of the harmonic ray th eorem if F and G are points at infinity the theorem should be interpre ted as assuming that the quadrilateral is a rectangle. Prove the theo em in this case (Maple is not required)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 290 "" 0 "" {TEXT -1 143 "EXERCISE: Interpret the harmonic \+ ray theorem when G is a point at infinity and F is not. Prove the theo rem in this case (Maple is not required)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 290 "" 0 "" {TEXT -1 189 "EXERCISE: Check the necessity \+ of the hypothesis that L be parallel to the line FG by doing the calcu lations for the harmonic ray theorem in a case when L is not parallel to the line FG. " }}}}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Table of Contents" 1 "vpstoc.mws" "" }{TEXT -1 0 "" }}}}{MARK "0 2 0 0" 2 } {VIEWOPTS 1 1 0 1 1 1803 }