GSP4g\QThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$5LEulerAndFive` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` ArialComic Sans MSqTimes New Roman    g'''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':' B"@@@@@@ @  @  ,?nQ\aAAn  UUUUUU?n  UUUUUU?n n"@u@@ @  @  wr?nQ\aACn         ?         F]tE?  F]tE?   Nj(Q?   ?nQ\aADn  n aT+?n n $?rV   Mf@|B @G@@]׎k@6;;f@@ !"@.hgU@(fnj@@ "$@ $! @&/@ '& @# @%/@ )#@ $),@!"$,"$! - -.I - (+C !* - 1,N 20> $> @F   &nQ\aA Animate Point> n!$"   (nQ\aAIsosceles Rightn<;f;f?F@\.B@?x -f f7<-RB< tտ[@\.R@@xt<t t?-RBi@\.R@@x> B$6"8!9    &nQ\aA ReSet Generic> n0     #&nQ\aA Show Objectsn  F]tE?n  F]tE?n =< n> 6<8?nQ\aAE n> ߿'&?n @? nA )w?n B?n ?Bn B?$?rVectsO CDEB F Cx;? G? BG>  GBD> @F @F   (nQ\aAAnimate Points > n$?"G!B   *nQ\aAIsosceles Generic> n8 =?B@<IHFCGEA>   z&nQ\aA Show Objects4n!"$  F]tE?/ N - O PN PN F]tE? Q (8E? SR T O? UR/ RQ - W W /9.˿ YR Z yZ? [R U[>  U[D> @F @F   (nQ\aAAnimate Points > n$U"[!R    &nQ\aA Right Generic> n4 PSUVTQR\[YZ]   )!&nQ\aA Show Objects nW /DxJzWٿnQ\aAF nW / n ba nc rN?n dan ea$LXz?N@ects da$LXz?N@ects gd> d$> @F    &nQ\aA Animate Point> n$d!a"g   &nQ\aA Equilateral > n( ghdeacbWf   &nQ\aA Show Objects> n!"$   &nQ\aA Show Vertices/n !%/n "&/n $# -n noO p2>  3q   >&nQ\aA Hide Segments> n02/p   ,nQ\aAShow CC, OC, CD, ICn 0p nt> n<uthgfedcba]\[ZYWVUTSRQPOIHGFEDCBA@?>=<9863qp20/    &&nQ\aA Hide Objectsn u) -n &w -n &w -n %w -n %w -n #w -n #w n   .nQ\aANine Point Circle--> n   6.nQ\aANine Point Circle-->6R n   .nQ\aANine Point Circle--> n   o.nQ\aANine Point Circle-->o> n4J^i   X :nQ\aA IMPORTANT!! ['Click' Me First!!]fn$$>nnQ\aAFzg@?2?I@@nQ\aAEZ/m@nQ2g@@nQ\aAD@}K@lؽ`Xm@@nQ\aAC@}K@lؽ`Xm@@nQ\aABZ/m@nQ2g@@nQ\aAAzg@?2?I@> @, 4;JL^`ikrv   ,nQ\aAShow Action Buttonsn>>> %VQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$ELNinePointCircle` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#`Comic Sans MS ArialSa g$    $ '''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':'      4nQ\aA<--Euler and Five Triangles _+n1>>>>>><0>><0>><0>>>>>1-fn > Znl@0@@v@u@@ @8@H@@ @  @ @ @ @   @ @   @/@ /@   -@ @  /@  /@ /@  -@ A       -  -  -  -  -  -  - ftt;J`>;Jl>    (nQ\aAShow Altitudesv> n   4nQ\aAShow Vertex-Ortho Segments> n    K6nQ\aAShow Perpendicular Bisectors> n     'BnQ\aA)Show Chords Connecting Midpoints of Sides> n    r <nQ\aA#Show Midpoints of Sides of Triangle> n   s,nQ\aAShow 9-Point Circle> n!$   4nQ\aAShow Feet of the AltitudesT> n   *nQ\aAShow OrthoCenterA> n #   @nQ\aA'Show MidPoints of Vertex-Ortho Segments > n    92nQ\aAShow Midpoints of ChordsO> n   ^6nQ\aAShow Center of 9-Point Circle> n(  #$ !    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The document will open as a copy.$a1LEuler Line (Algebra)` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $    g'''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':') AF@u@4(@ 4p\.1@: &@  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n   On^ d<0> >02?YY@??nQ\aAad6 On^ <0> >02?YY@??nQ\aAb5 On^ <0> >02?YY@?? nQ\aAcX30 3nb g8<0> B04!4!MDwCXwSwr`0 3nb gM<0> B0&&MDwCXwSwr`0 3n b gb<0> B0))MDwCXwSwr` On^ <0> >0?YY@??nQ\aAt[1]%E8n BE8 C    /   /  / ,, - I - C  - N/  -  O ! pp?\<+><+>>>>>>?||h<+><+><+>>>>>76>76Xshh:yT>a:y^> #"!   4c<nQ\aA#Show Euler Line (generated by tool)0 3n  9<0>R v0&&MDwCXwSwr ````  ```` 0 3  <0>T2n  0# MDwCXwSwr `````  ```` G 9 )*f W<0>T2sl F0-&-&MDwCXwSwr``fH: +  uc:ҒL,G@Xeate> ,   R7LnQ\aA2Show Function Plotted Using Algebraic Calculations n   ,nQ\aA<<--9 Point Circlee`  n   c0nQ\aAEuler Line (Proof)-->>sc n``:L>>>>:n5>>>>>>>>>>>>>>5-#VQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. 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The document will open as a copy.$>LEuler Line (Centroid)` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $   '''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':') OF@u@4(@ 4p\.1@: &@  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n  npp\<+><+>>>>>>n||:h<+><+><+>>>>>:n>W On^ <0>H<S# >02?YY@??nQ\aAa & On^ *<0>H<S# >02?YY@??nQ\aAb*!= On^ =<0>H<S# >02?YY@?? nQ\aAc=!P0 3n  <0> v0&&MDwCXwSwr ````  ```` 0 3  @<0>T2n  0# MDwCXwSwr `````  ```` G 9  f v<0>0R    KLnQ\aA2Show Function Plotted Using Algebraic Calculations0 3n b a<0>0R0R0RH<S# >0?YY@??nQ\aAt[1]F%E8nB /  /  - C /  - !O,, - #$I  & ' - ( N ) ")> *)+"   N<nQ\aA#Show Euler Line (generated by tool)nXXE>>^n$r N (2b , 2c)`i@[@ @ &@$ M (2a , 0) /. . / 0& P (a+b , c )n 2$ R (a , 0 )) 3 .488>)%>>>)I    2nQ\aA<<--Euler Line (Algebra)d nLL7M8><(>>>>7M|ndd:Q><(>>>>>:nLJA7><(>><(>>>AnHH5><(>><(>>>nDz><(>><(>><(>><(>>>>>1x2D6" n   6nQ\aAEuler Line (CircumCenter)-->>4"-#VQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. 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The document will open as a copy.$'@LEuler Line (CircumCenter)` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $   '''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':') OF@u@4(@ 4p\.1@: &@  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n  npp\<+><+>>>>>>n||:h<+><+><+>>>>>:n>T On^ <0>2U10 >02?YY@??nQ\aAa & On^ *<0>2U10 >02?YY@??nQ\aAb*!= On^ =<0>2U10 >02?YY@?? nQ\aAc=!P0 3n  <0>R v0&&MDwCXwSwr ````  ```` 0 3  @<0>T2n  0# MDwCXwSwr `````  ```` G 9  f v<0>0R    KLnQ\aA2Show Function Plotted Using Algebraic Calculations0 3n b a<0>0R0R0R2U10 >0?YY@??nQ\aAt[1]F%E8nB /  /  - C /  - !O,, - #$I  & ' - ( N ) ")> *)+"   N<nQ\aA#Show Euler Line (generated by tool)nXXE>>fn$ N (2b , 2c)`i@[@ @ &@$ M (2a , 0) /. . / 0& P (a+b , c )n 2$ R (a , 0 )) 3 .4((3(>3(H    v2nQ\aA<<--Euler Line (Centroid)v  nXX,LE>>>>>>,Lcn/>>><(>>><+><+>>>>>/Bn0.:>>>:enllY><(><+>>>>>>/n9><(>><+>>>>>>>9#/n 42/n 30 n   x6nQ\aAEuler Line (OrthoCenter)-->> x!"-#VQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. 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The document will open as a copy.$BLEuler Line (OrthoCenter)` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $   '''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':') OF@u@4(@ 4p\.1@: &@  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n  npp\<+><+>>>>>>n||:h<+><+><+>>>>>:n>)V On^ <0>H<!5 >02?YY@??nQ\aAa & On^ *<0>H<!5 >02?YY@??nQ\aAb*!= On^ =<0>H<!5 >02?YY@?? nQ\aAc=!P0 3n  <0>H<R1 v0&&MDwCXwSwr ````  ```` 0 3  @<0>T2n  0# MDwCXwSwr `````  ```` G 9  f v<0>0R    KLnQ\aA2Show Function Plotted Using Algebraic Calculations0 3n b a<0>0R0R0RH<!5 >0?YY@??nQ\aAt[1]F%E8nB /  /  - C /  - !O,, - #$I  & ' - ( N ) ")> *)+"   N<nQ\aA#Show Euler Line (generated by tool)nXXE>>qn$ N (2b , 2c)`i@[@ @ &@$ M (2a , 0) /. . / 0& P (a+b , c )n 2$ R (a , 0 )) 3 .4((O(>=O(H    6nQ\aA<<--Euler Line (CircumCenter)E n\ZHLF>>>>>>?HLcnllGoX>>><(>>>>>Gon00R>>>RnXXD><(<+>>>>>>>=nUm><(><+>>>>>>>U#/n 42/n 30/n .2/n 0 n   0nQ\aAEquation of Euler Linee!-#VQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. 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The document will open as a copy.$RLEquation of The Euler Line` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $    g'''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':') OF@u@4(@ 4p\.1@: &@  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n  npp\<+><+>>>>>>n||:h<+><+><+>>>>>>:n>T On^ <0>H<xy >02?YY@??nQ\aAa & On^ *<0>H<xy >02?YY@??nQ\aAb*!= On^ =<0>H<xy >02?YY@?? nQ\aAc=!P0 3n  <0>H<R1 v0&&MDwCXwSwr ````  ```` 0 3  @<0>T2n  0# MDwCXwSwr `````  ```` G 9  f v<0>0R    KLnQ\aA2Show Function Plotted Using Algebraic Calculations0 3n b a<0>0R0R0RH<xy >0?YY@??nQ\aAt[1]F%E8nB /  /  - C /  - !O,, - #$I  & ' - ( N ) ")> *)+"   N<nQ\aA#Show Euler Line (generated by tool)nXXE>> n$r N (2b , 2c)`i@[@ @ &@$ M (2a , 0) /. . / 0& P (a+b , c )n 2$ R (a , 0 )) 3 .400/>/O    ;6nQ\aA<<--Euler Line (OrthoCenter) ;nWm><(><+>>>>>>>WH/n 42/n 30/n .2/n 0 n1/n >1,n/.,/. - @AI - ?:C - 56N/ /1 - E<O FD DC    ?2nQ\aA-->>Back to Beginning-->> ?0nZ><(>><+>>>>>>>Z|n  >><->>><->>>>>>> )fnw> n||? h<+><+><+>>>>>>? @nppE\<+><+>>>>>>Efnxvwb>c>> w]nl<(<+><+>>>>><0>>Intta<+><+><+>>>>>I;n PQ<(<+><+>>>>><0>>><+><+><+>>>>>>>-#\{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LCircumcenter Arial  nQ\aACnQ\aABnQ\aAAv@@ @  @nQ\aAD nnQ\aAE/n nQ\aAj/n nQ\aAk -n  C\{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LCentroid Arial  nQ\aACnQ\aABnQ\aAA   nQ\aAD nnQ\aAEn n  -n  N\{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LOrthoCenter Arial  nQ\aACnQ\aABnQ\aAA  / nQ\aAj/n nQ\aAk -n O\{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LIncenter Arial  nQ\aAC0Pq@ܾ BW@@nQ\aABnQ\aAA,nQ\aAj,nnQ\aAk -n I\{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LEulerPlusIncenter Arial  enQ\aACe@@nQ\aABnQ\aAA    nQ\aAD nnQ\aAE/n nQ\aAj nnQ\aAFn /n nQ\aAk,nnQ\aAl,nnQ\aAm -n  I/  nQ\aAn -n  C  -  N/ nQ\aAo -n  O