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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Steiners Problem" }}{PARA 0 "" 0 "
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{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(linalg
); with(plots);" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected
 names norm and trace have been redefined and unprotected\n" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlo
ckG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjoint
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vecpotentG%(vectdimG%'vectorG%*wronskianG" }}{PARA 7 "" 1 "" {TEXT -1 
50 "Warning, the name changecoords has been redefined\n" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#7W%(animateG%*animate3dG%-animatecurveG%&arrowG%
-changecoordsG%,complexplotG%.complexplot3dG%*conformalG%,conformal3dG
%,contourplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplot
G%,densityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradp
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ions3dG%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textp
lotG%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "a:=15; b:=1+2*(-1)^(8+6);c:=13+6+((1+6)/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"aG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG#\"#X\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "AM:=sqrt((x+a)^2+y^2); BM:=sqrt((x-
a)^2+y^2); CM:=sqrt((x-b)^2+(y-c)^2);" }{TEXT -1 2 "  " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#AMG*$-%%sqrtG6#,**$)%\"xG\"\"#\"\"\"F.*&\"#IF.F
,F.F.\"$D#F.*$)%\"yGF-F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BMG
*$-%%sqrtG6#,**$)%\"xG\"\"#\"\"\"F.*&\"#IF.F,F.!\"\"\"$D#F.*$)%\"yGF-F
.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CMG,$*$-%%sqrtG6#,,*$)%\"x
G\"\"#\"\"\"\"\"%*&\"#CF/F-F/!\"\"\"%h?F/*&F0F/)%\"yGF.F/F/*&\"$!=F/F7
F/F3F/#F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=AM+BM+CM;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*$-%%sqrtG6#,**$)%\"xG\"\"
#\"\"\"F/*&\"#IF/F-F/F/\"$D#F/*$)%\"yGF.F/F/F/F/*$-F(6#,*F+F/*&F1F/F-F
/!\"\"F2F/F3F/F/F/*&#F/F.F/-F(6#,,F+\"\"%*&\"#CF/F-F/F;\"%h?F/*&FAF/F4
F/F/*&\"$!=F/F5F/F;F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
fx:=diff(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,(*&,**$)%\"xG
\"\"#\"\"\"F,*&\"#IF,F*F,F,\"$D#F,*$)%\"yGF+F,F,#!\"\"F+,&F*F+F.F,F,#F
,F+*(F6F,,*F(F,*&F.F,F*F,F4F/F,F0F,F3,&F*F+F.F4F,F,*(#F,\"\"%F,,,F(F=*
&\"#CF,F*F,F4\"%h?F,*&F=F,F1F,F,*&\"$!=F,F2F,F4F3,&F*\"\")F@F4F,F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fy:=diff(f,y);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#fyG,(*&,**$)%\"xG\"\"#\"\"\"F,*&\"#IF,F*F,F,
\"$D#F,*$)%\"yGF+F,F,#!\"\"F+F2F,F,*&,*F(F,*&F.F,F*F,F4F/F,F0F,F3F2F,F
,*(#F,\"\"%F,,,F(F:*&\"#CF,F*F,F4\"%h?F,*&F:F,F1F,F,*&\"$!=F,F2F,F4F3,
&F2\"\")FAF4F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fxx:=di
ff(fx,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$fxxG,.*&,**$)%\"xG\"\"
#\"\"\"F,*&\"#IF,F*F,F,\"$D#F,*$)%\"yGF+F,F,#!\"$F+,&F*F+F.F,F+#!\"\"
\"\"%*&F,F,*$-%%sqrtG6#F'F,F7F,*&#F,F8F,*&,*F(F,*&F.F,F*F,F7F/F,F0F,F3
,&F*F+F.F7F+F,F7*&F,F,*$-F<6#FAF,F7F,*&#F,\"\")F,*&,,F(F8*&\"#CF,F*F,F
7\"%h?F,*&F8F,F1F,F,*&\"$!=F,F2F,F7F3,&F*FJFNF7F+F,F7*&F+F,FL#F7F+F," 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fyy:=diff(fy,y);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$fyyG,.*&,**$)%\"xG\"\"#\"\"\"F,*&\"
#IF,F*F,F,\"$D#F,*$)%\"yGF+F,F,#!\"$F+F2F+!\"\"*&F,F,*$-%%sqrtG6#F'F,F
5F,*&,*F(F,*&F.F,F*F,F5F/F,F0F,F3F2F+F5*&F,F,*$-F96#F<F,F5F,*&#F,\"\")
F,*&,,F(\"\"%*&\"#CF,F*F,F5\"%h?F,*&FGF,F1F,F,*&\"$!=F,F2F,F5F3,&F2FDF
MF5F+F,F5*&F+F,FF#F5F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"fxy:=diff(fx,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fxyG,(*(,**$)%
\"xG\"\"#\"\"\"F,*&\"#IF,F*F,F,\"$D#F,*$)%\"yGF+F,F,#!\"$F+,&F*F+F.F,F
,F2F,#!\"\"F+*&#F,F+F,*(,*F(F,*&F.F,F*F,F7F/F,F0F,F3,&F*F+F.F7F,F2F,F,
F7*&#F,\"\")F,*(,,F(\"\"%*&\"#CF,F*F,F7\"%h?F,*&FCF,F1F,F,*&\"$!=F,F2F
,F7F3,&F*F@FEF7F,,&F2F@FIF7F,F,F7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "svar:=solve(\{fx,fy\},\{x,y\});" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%svarG<$/%\"xG,&#!$5#\"#P\"\"\"*&#\"$5(F+F,)-%'RootOf
G6$,(\"%hZF,*&\"&-d\"F,)%#_ZG\"\"#F,!\"\"*&\"%T)*F,)F9\"\"%F,F,$\"+*H,
,Q'!#5F:F,F,/%\"yG,&*$F0F,#!$f%\"$[\"#\"%\\9FIF," }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 49 "u:=evalf(rhs(svar[1]));  v:=evalf(rhs(svar[2]
)); " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 124 "                                      Koordinaterna ti
ll den s\366kta extrempunkten som ger den totalt kortaste v\344gen AM+
BM+CM!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+'o;a8#!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG$\"+%*R6G&)!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A:=eval(fxx,[x=u,y=v]); B:=eval(fxy
,[x=u,y=v]); C:=eval(fyy,[x=u,y=v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG$\"+2)[J,\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG$\"+CM
(Qf%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG$\"+^Sh:()!#6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Ma:=matrix(2,2,[A,B,B,C]); E
gv:=eigenvalues(Ma);" }}{PARA 11 "" 0 "" {TEXT -1 27 "Egenv\344rden, b
\345da positiva!!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MaG-%'matrixG6
#7$7$$\"+2)[J,\"!#5$\"+CM(Qf%!#77$F-$\"+^Sh:()!#6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$EgvG6$$\"+'zC'z&)!#6$\"+KxuE5!#5" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "fM:=eval(f,[x=u,y=v]);  " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "fA:=eval(f,[x=-a,y=0]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "fB:=eval(f,[x=a,y=0]);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "fC:=eval(f,[x=b,y=c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f
MG$\"+u$\\t&[!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fAG,&*$-%%sqrt
G6#\"$+*\"\"\"F+*&#F+\"\"#F+-F(6#\"%@LF+F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#fBG,&*$-%%sqrtG6#\"$+*\"\"\"F+*&#F+\"\"#F+-F(6#\"%,E
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\"-F(6#\"\"%F+#F+F.*(F/F+-F(6#\"%,EF+F,F+F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "Av:=vector([-a,0]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#AvG-%'vectorG6#7$!#:\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "Bv:=vector([a,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#BvG-%'vectorG6#7$\"#:\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "Cv:=vector([b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CvG-%
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18 "Mv:=vector([u,v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MvG-%'vec
torG6#7$$\"+'o;a8#!\"*$\"+%*R6G&)F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "MA:=normalize(Av-Mv); MB:=normalize(Bv-Mv); MC:=norma
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+^6`_*)!#5$!++#zbX%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MBG-%'vect
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0 "" {MPLTEXT 1 0 62 "sk1:=dotprod(MA,MB); sk2:=dotprod(MA,MC); sk3:=d
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{XPPMATH 20 "6#>%$sk1G$!+++++]!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$sk2G$!+********\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sk3G$!+/+
++]!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MMAG$\"+++++5!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MMCG$\"+++++5!\"*" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$MMBG$\"+)*********!#5" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 89 "vink1:=arccos(sk1/(MMA*MMB)); vink2:=arccos(sk2/(MM
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"" {TEXT -1 150 "                                                     \+
                                             Vinklarna mellan vektorer
na. totala summan=2*pi, ok!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&vink
1G$\"+.^R%4#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&vink2G$\"+-^R%4
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{XPPMATH 20 "6#>%&list1G7,7$$\"+'o;a8#!\"*$\"+%*R6G&)F)7$!#:\"\"!7$\"#
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