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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Module 2 : Geometry" }}{PARA 3 "
" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 27 "204 : Centers Of A
 Triangle" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
1029 "What is the center of a triangle? Well there are at least four d
ifferent answers :\n\011 the incenter is the center of the inscribed c
ircle for a triangle\n\011 the circumcenter is the center of the circu
mscribed circle for a triangle\n\011 the centroid is the center of mas
s for a triangle.\n\011 the orthocenter is the center of the intersect
ion of the altitudes\nIts a fascinating fact that each of these center
s is a point of concurrency of a triangle. Any two non-parallel lines \+
will intersect. However, if you have three lines, it is highly unlikel
y they would happen to intersect at the same point. Such a point is a \+
point of concurrency for the three lines. Each of the four points ment
ioned above is the intersection of three easily constructed lines for \+
the triangle.\n\nIn this module, we will examine each of these points,
 and how they are defined, and see what they look like when plotted on
 a triangle. At the conclusion of this module you will better understa
nd what each of the centers of a triangle is and how they are construc
ted." }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________
__________________________________________________" }}{PARA 4 "" 0 "" 
{TEXT -1 8 "A. Setup" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________
__________________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "We will \+
begin by invoking the geometry spirits and defining some options which
 will save us typing later. If you return to this project at a later d
ate, be sure to re-execute these definitions. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " with(geomet
ry):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " fc := 'filled = \+
true, color = coral':\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
 fbk := 'filled = true, color = black':\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 " cr := 'color = red, thickness = 2':\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " cb := 'color = blue, thickness = 2
':\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " cg := 'color = gre
en, thickness = 2':\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " c
w := 'color = white, thickness = 2':" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 79 "We are going to define a single trian
gle and then do all kinds of things to it." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " point( A, [-3,-2]
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"_____________________________________________________________________
______________" }}{PARA 4 "" 0 "" {TEXT -1 34 "B. Angle Bisectors   ->
   Incenter" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________
_________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "A triangle has three a
ngles. Each angle can be bisected by a line. The intersection of these
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"" 0 "" {TEXT -1 44 "C. Perpindicular Bisectors  ->  Circumcenter" }}
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__________________________________________" }}{PARA 0 "" 0 "" {TEXT 
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400 "A perpendicular bisector of a segment is another line which bisec
ts the segment and at the same time is perpendicular to the segment. E
ach side of a triangle is a line segment, and therefore can be bisecte
d. When all the perpendicular bisectors are drawn for all three sides,
 they intersect in a single point called the circumcenter which is the
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" {TEXT -1 24 "D. Medians  ->  Centroid" }}{PARA 0 "" 0 "" {TEXT -1 
83 "__________________________________________________________________
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 536 "A median of a triangle i
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e. Note that a median, is usually not an angle bisector nor a perpendi
cular bisector, except in certain special cases. The point where the t
hree medians meet is called the centroid. This is a very special point
 because it represents the center of gravity of the triangle. If the t
riangle were made out of metal or wood of even thickness, the centroid
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ip of a single finger!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
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+Z*y:j&FP7$$\"++++S=FJ$\"+;j_5iFP7$$\"++++?;FJ$\"+%ot%*y'FP7$$\"+++++9
FJ$\"+`5UotFP7$$\"++++!=\"FJ$\"+@%ot%zFP7$$\"*+++g*FJ$\"+*y:j_)FP7$$\"
*+++S(FJ$\"+eJE0\"*FP7$$\"*+++?&FJ$\"+E0@%o*FP7$$\"*++++$FJ$\"+*y:j-\"
FJ7$$\")+++!)FJ$\"+E0@%3\"FJ7$$!*+++S\"FJ$\"+j_5U6FJ7$$!*+++g$FJ$\"+++
++7FJ7$$!*+++!eFJ$\"+PZ*yD\"FJ7$$!*++++)FJ$\"+u%*y:8FJ7$$!++++?5FJ$\"+
6Uot8FJ7$$!++++S7FJ$\"+Z*y:V\"FJ7$$!++++g9FJ$\"+%ot%*[\"FJ7$$!++++!o\"
FJ$\"+@%ota\"FJ7$$!+++++>FJ$\"+eJE0;FJ7$$!*+++7#F@$\"+&*y:j;FJ7$$!*+++
M#F@$\"+KE0@<FJ7$$!*+++c#F@$\"+ot%*y<FJ7$$!*+++y#F@$\"+0@%o$=FJ7$$!*++
++$F@$\"+Uot%*=FJ-F76#%%LINEG-F;6&F=F>F3F3-FD6%7U7$F.$!++++!e\"F@7$FL$
!+++!G_\"F@7$FR$!+++gl9F@7$FW$!+++S39F@7$Ffn$!+++?^8F@7$F[o$!++++%H\"F
@7$F`o$!+++!oB\"F@7$Feo$!+++gz6F@7$Fjo$!+++SA6F@7$F_p$!+++?l5F@7$Fdp$!
++++35F@7$Fip$!++++3&*FJ7$F^q$!++++O*)FJ7$Fcq$!++++k$)FJ7$Fhq$!++++#z(
FJ7$F]r$!++++?sFJ7$Fbr$!++++[mFJ7$Fhr$!++++wgFJ7$F]s$!++++/bFJ7$Fbs$!+
+++K\\FJ7$Fgs$!++++gVFJ7$F\\t$!++++)y$FJ7$Fat$!++++;KFJ7$Fft$!++++WEFJ
7$F[u$!++++s?FJ7$F`u$!+++++:FJ7$Feu$!++++!G*FP7$Fju$!++++gNFP7$F_v$\"+
+++g@FP7$Fdv$\"++++!)yFP7$Fiv$\"++++g8FJ7$F^w$\"++++K>FJ7$Fcw$\"++++/D
FJ7$Fhw$\"++++wIFJ7$F]x$\"++++[OFJ7$Fbx$\"++++?UFJ7$Fgx$\"++++#z%FJ7$F
\\y$\"++++k`FJ7$Fay$\"++++OfFJ7$Ffy$\"++++3lFJ7$F[z$\"++++!3(FJ7$F`z$
\"++++_wFJ7$Fez$\"++++C#)FJ7$Fjz$\"++++'z)FJ7$F_[l$\"++++o$*FJ7$Fd[l$
\"++++S**FJ7$Fi[l$\"+++?^5F@7$F^\\l$\"+++S36F@7$Fc\\l$\"+++gl6F@7$Fh\\
l$\"+++!GA\"F@7$F]]l$\"++++!G\"F@Fa]lFd]l-FD6%7U7$F.$\"+'G9dG%FJ7$FL$
\"++++gTFJ7$FR$\"+9dGMSFJ7$FW$\"+H9d3RFJ7$Ffn$\"+Vr&Gy$FJ7$F[o$\"+dG9d
OFJ7$F`o$\"+r&G9`$FJ7$Feo$\"+'G9dS$FJ7$Fjo$\"++++!G$FJ7$F_p$\"+9dGaJFJ
7$Fdp$\"+H9dGIFJ7$Fip$\"+Vr&G!HFJ7$F^q$\"+dG9xFFJ7$Fcq$\"+r&G9l#FJ7$Fh
q$\"+'G9d_#FJ7$F]r$\"+++++CFJ7$Fbr$\"+9dGuAFJ7$Fhr$\"+H9d[@FJ7$F]s$\"+
Vr&G-#FJ7$Fbs$\"+dG9(*=FJ7$Fgs$\"+r&G9x\"FJ7$F\\t$\"+'G9dk\"FJ7$Fat$\"
++++?:FJ7$Fft$\"+9dG%R\"FJ7$F[u$\"+H9do7FJ7$F`u$\"+Vr&G9\"FJ7$Feu$\"+d
G9<5FJ7$Fju$\"+9dG9*)FP7$F_v$\"+dG9dwFP7$Fdv$\"+++++kFP7$Fiv$\"+Vr&G9&
FP7$F^w$\"+'G9d)QFP7$Fcw$\"+H9dGEFP7$Fhw$\"+r&G9P\"FP7$F]x$Fb\\mFfr7$F
bx$!+Vr&G9\"FP7$Fgx$!+++++CFP7$F\\y$!+dG9dOFP7$Fay$!+9dG9\\FP7$Ffy$!+r
&G9<'FP7$F[z$!+H9dGuFP7$F`z$!+'G9do)FP7$Fez$!+Vr&G%**FP7$Fjz$!++++?6FJ
7$F_[l$!+'G9dC\"FJ7$Fd[l$!+r&G9P\"FJ7$Fi[l$!+dG9(\\\"FJ7$F^\\l$!+Vr&Gi
\"FJ7$Fc\\l$!+H9d[<FJ7$Fh\\l$!+9dGu=FJ7$F]]l$!+++++?FJFa]lFd]l-%(SCALI
NGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F(F.;F+F4" 1 2 0 1 
10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C
urve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 248 "The centroid has another special property. It cuts each \+
median into two pieces, one twice as long as the other. We can verify \+
this. Note that the distance from the centroid to each vertex is exact
ly twice the distance from the centroid to midpoint." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 " centroid(
 ct, T):  midpoint( mAB, A, B):  midpoint( mBC, B, C):  midpoint( mAC,
 A, C):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " evalf( distan
ce( C, ct)) ;   \011evalf( distance( mAB, ct)) ;  \n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"+$4'zUY!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+Y!)R@B!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " evalf( di
stance( A, ct)) ;    \011evalf( distance( mBC, ct)) ;  \n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"+*\\Q[P&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+\\#>uo#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " e
valf( distance( B, ct)) ; \011\011evalf( distance( mAC, ct)) ;  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+**3'*[l!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+^/[uK!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 83 "_______________________________________________
____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 29 "E
. Altitudes  ->  Orthocenter" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 438 "An altitude is a line segment that ext
ends from one vertex to the opposite side, or an extension of that opp
osite side, and is perpendicular to it. An altitude is somewhat like a
 median in that it goes from vertex to opposite side, however, it does
 not necessarily pass through the midpoint of the opposite side. Many \+
times, the altitude is located outside of the circle. The intersection
 of the three altitudes is called the orthocenter." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " altitude(al
t1,A,T):  altitude(alt2,B,T):  altitude(alt3,C,T):\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 " draw( \{T(fc), alt1,alt2, alt3 \});" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%)POLYGONSG6%7%
7$$!\"$\"\"!$!\"#F*7$$\"\")F*$!\"\"F*7$$F*F*$\"\"&F*-%&STYLEG6#%,PATCH
NOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\F@F3-%'CURVESG6%7U7$F
.$\"+nmmm7F@7$$\"++++!y(!\"*$\"+LLLP7F@7$$\"++++gvFM$\"++++37F@7$$\"++
++StFM$\"+nmmy6F@7$$\"++++?rFM$\"+LLL\\6F@7$$\"+++++pFM$\"++++?6F@7$$
\"++++!o'FM$\"+nmm!4\"F@7$$\"++++gkFM$\"+LLLh5F@7$$\"++++SiFM$\"++++K5
F@7$$\"++++?gFM$\"+nmm-5F@7$$\"+++++eFM$\"+LLLL(*FM7$$\"++++!e&FM$\"++
++S%*FM7$$\"++++g`FM$\"+nmmY\"*FM7$$\"++++S^FM$\"+LLL`))FM7$$\"++++?\\
FM$\"++++g&)FM7$$\"+++++ZFM$\"+nmmm#)FM7$$\"++++![%FM$\"+LLLtzFM7$$\"+
+++gUFM$\"++++!o(FM7$$\"++++SSFM$\"+nmm'Q(FM7$$\"++++?QFM$\"+LLL$4(FM7
$$\"+++++OFM$\"+++++oFM7$$\"++++!Q$FM$\"+nmm1lFM7$$\"++++gJFM$\"+LLL8i
FM7$$\"++++SHFM$\"++++?fFM7$$\"++++?FFM$\"+nmmEcFM7$$\"+++++DFM$\"+LLL
L`FM7$$\"++++!G#FM$\"++++S]FM7$$\"++++g?FM$\"+nmmYZFM7$$\"++++S=FM$\"+
LLL`WFM7$$\"++++?;FM$\"++++gTFM7$$\"+++++9FM$\"+nmmmQFM7$$\"++++!=\"FM
$\"+LLLtNFM7$$\"*+++g*FM$\"++++!G$FM7$$\"*+++S(FM$\"+nmm')HFM7$$\"*+++
?&FM$\"+LLL$p#FM7$$\"*++++$FM$\"+++++CFM7$$\")+++!)FM$\"+nmm1@FM7$$!*+
++S\"FM$\"+LLL8=FM7$$!*+++g$FM$\"++++?:FM7$$!*+++!eFM$\"+nmmE7FM7$$!*+
+++)FM$\"+LLLL$*!#57$$!++++?5FM$\"+++++kF]z7$$!++++S7FM$\"+nmmmMF]z7$$
!++++g9FM$Fau!#67$$!++++!o\"FM$!+++++CF]z7$$!+++++>FM$!+LLLL`F]z7$$!*+
++7#F@$!+nmmm#)F]z7$$!*+++M#F@$!++++?6FM7$$!*+++c#F@$!+LLL89FM7$$!*+++
y#F@$!+nmm1<FM7$$!*++++$F@$!+++++?FM-F76#%%LINEG-F;6&F=F>F3F3-FD6%7U7$
F.$!+++++5FM7$FK$!+dG9d!*F]z7$FQ$!+9dG9\")F]z7$FV$!+r&G9<(F]z7$Fen$!+H
9dGiF]z7$Fjn$!+'G9dG&F]z7$F_o$!+Vr&GM%F]z7$Fdo$!+++++MF]z7$Fio$!+dG9dC
F]z7$F^p$!+9dG9:F]z7$Fcp$!+9dG9dF\\[l7$Fhp$\"+9dG9PF\\[l7$F]q$\"+9dG98
F]z7$Fbq$\"+dG9dAF]z7$Fgq$\"+++++KF]z7$F\\r$\"+Vr&G9%F]z7$Far$\"+'G9d3
&F]z7$Ffr$\"+H9dGgF]z7$F[s$\"+r&G9(pF]z7$F`s$\"+9dG9zF]z7$Fes$\"+dG9d)
)F]z7$Fjs$\"+++++)*F]z7$F_t$\"+9dGu5FM7$Fdt$\"+H9do6FM7$Fit$\"+Vr&GE\"
FM7$F^u$\"+dG9d8FM7$Fcu$\"+r&G9X\"FM7$Fhu$\"+'G9da\"FM7$F]v$\"++++S;FM
7$Fbv$\"+9dGM<FM7$Fgv$\"+H9dG=FM7$F\\w$\"+Vr&G#>FM7$Faw$\"+dG9<?FM7$Ff
w$\"+r&G96#FM7$F[x$\"+'G9d?#FM7$F`x$\"+++++BFM7$Fex$\"+9dG%R#FM7$Fjx$
\"+H9d)[#FM7$F_y$\"+Vr&Ge#FM7$Fdy$\"+dG9xEFM7$Fiy$\"+r&G9x#FM7$F_z$\"+
'G9d'GFM7$Fdz$\"++++gHFM7$Fiz$\"+9dGaIFM7$F^[l$\"+H9d[JFM7$Fc[l$\"+Vr&
GC$FM7$Fh[l$\"+dG9PLFM7$F]\\l$\"+r&G9V$FM7$Fb\\l$\"+'G9d_$FM7$Fg\\l$\"
++++?OFM7$F\\]l$F[`lFMF`]lFc]l-FD6%7U7$F.$!+++++$)F@7$FK$!++++e!)F@7$F
Q$!++++;yF@7$FV$!++++uvF@7$Fen$!++++KtF@7$Fjn$!++++!4(F@7$F_o$!++++[oF
@7$Fdo$!++++1mF@7$Fio$!++++kjF@7$F^p$!++++AhF@7$Fcp$!++++!)eF@7$Fhp$!+
+++QcF@7$F]q$!++++'R&F@7$Fbq$!++++a^F@7$Fgq$!++++7\\F@7$F\\r$!++++qYF@
7$Far$!++++GWF@7$Ffr$!++++'=%F@7$F[s$!++++WRF@7$F`s$!++++-PF@7$Fes$!++
++gMF@7$Fjs$!++++=KF@7$F_t$!++++wHF@7$Fdt$!++++MFF@7$Fit$!++++#\\#F@7$
F^u$!++++]AF@7$Fcu$!++++3?F@7$Fhu$!++++m<F@7$F]v$!++++C:F@7$Fbv$!++++#
G\"F@7$Fgv$!++++S5F@7$F\\w$!++++!)zFM7$Faw$!++++gbFM7$Ffw$!++++SJFM7$F
[x$!+++++sF]z7$F`x$\"+++++<FM7$Fex$\"++++?TFM7$Fjx$\"++++SlFM7$F_y$\"+
+++g*)FM7$Fdy$\"++++Q6F@7$Fiy$\"++++!Q\"F@7$F_z$\"++++A;F@7$Fdz$\"++++
k=F@7$Fiz$\"++++1@F@7$F^[l$\"++++[BF@7$Fc[l$\"++++!f#F@7$Fh[l$\"++++KG
F@7$F]\\l$\"++++uIF@7$Fb\\l$\"++++;LF@7$Fg\\l$\"++++eNF@7$F\\]l$\"++++
+QF@F`]lFc]l-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$
;F(F.;F+F4" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 4 "" 0 "" {TEXT -1 19 "F. All Together Row" }}{PARA 0 "
" 0 "" {TEXT -1 83 "__________________________________________________
_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "We've se
en all four points of concurrency. How do they look together? Lets dra
w all 12 of the lines discussed on a single diagram." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "draw( \{ \+
p1(cb),p2(cb),p3(cb), bi1(cr),bi2(cr),bi3(cr),\n\011md1(cg), md2(cg), \+
md3(cg), alt1(cw), alt2(cw), alt3(cw), T(fbk) \});" }}{PARA 13 "" 1 "
" {GLPLOT2D 400 300 300 {PLOTDATA 2 "62-%)POLYGONSG6%7%7$$!\"$\"\"!$!
\"#F*7$$\"\")F*$!\"\"F*7$$F*F*$\"\"&F*-%&STYLEG6#%,PATCHNOGRIDG-%'COLO
URG6&%$RGBGF*F*F*-%'CURVESG6&7U7$F.$!+PVz#\\#!\")7$$\"++++!y(!\"*$!+$
\\#\\5CFE7$$\"++++gvFI$!+[1>GBFE7$$\"++++StFI$!+/)))eC#FE7$$\"++++?rFI
$!+gpej@FE7$$\"+++++pFI$!+;^G\"3#FE7$$\"++++!o'FI$!+rK)*)*>FE7$$\"++++
gkFI$!+F9o;>FE7$$\"++++SiFI$!+$ezV$=FE7$$\"++++?gFI$!+Rx2_<FE7$$\"++++
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+++S^FI$!+i.(GU\"FE7$$\"++++?\\FI$!+<&o0M\"FE7$$\"+++++ZFI$!+tmEe7FE7$
$\"++++![%FI$!+H['f<\"FE7$$\"++++gUFI$!+%)Hm$4\"FE7$$\"++++SSFI$!+S6O6
5FE7$$\"++++?QFI$!+iHf!H*FI7$$\"+++++OFI$!+>Xdn%)FI7$$\"++++!Q$FI$!+wg
bWwFI7$$\"++++gJFI$!+Lw`@oFI7$$\"++++SHFI$!+!>>&)*fFI7$$\"++++?FFI$!+Z
2]v^FI7$$\"+++++DFI$!+/B[_VFI7$$\"++++!G#FI$!+hQYHNFI7$$\"++++g?FI$!+>
aW1FFI7$$\"++++S=FI$!+wpU$)=FI7$$\"++++?;FI$!+L&3/1\"FI7$$\"+++++9FI$!
+)*3!RP#!#57$$\"++++!=\"FI$\"+JNGceFgv7$$\"*+++g*FI$\"+$zY'39FI7$$\"*+
++S(FI$\"+P_mJAFI7$$\"*+++?&FI$\"+zOoaIFI7$$\"*++++$FI$\"+@@qxQFI7$$\"
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FI7$$!++++S7FI$\"+A7$)Q'*FI7$$!++++g9FI$\"+m\\=Y5FE7$$!++++!o\"FI$\"+5
o[G6FE7$$!+++++>FI$\"+b')y57FE7$$!*+++7#FE$\"+*\\!4$H\"FE7$$!*+++M#FE$
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$\"+wyHA;FE-F76#%%LINEG-F;6&F=$\"*++++\"FEF3F3-%*THICKNESSG6#\"\"#-F?6
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\"++++?:FI7$Fay$\"+nmmE7FI7$Ffy$\"+LLLL$*Fgv7$F[z$\"+++++kFgv7$F`z$\"+
nmmmMFgv7$Fez$Fgbl!#67$Fjz$!+++++CFgv7$F_[l$!+LLLL`Fgv7$Fd[l$!+nmmm#)F
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\\]l-F;6&F=\"\"\"FeglFeglFc]l-F?6&7U7$F.$!+++++5FI7$FG$!+dG9d!*Fgv7$FM
$!+9dG9\")Fgv7$FR$!+r&G9<(Fgv7$FW$!+H9dGiFgv7$Ffn$!+'G9dG&Fgv7$F[o$!+V
r&GM%Fgv7$F`o$!+++++MFgv7$Feo$!+dG9dCFgv7$Fjo$!+9dG9:Fgv7$F_p$!+9dG9dF
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gv7$Fhq$\"+Vr&G9%Fgv7$F]r$\"+'G9d3&Fgv7$Fbr$\"+H9dGgFgv7$Fgr$\"+r&G9(p
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\"+++++BFI7$Fbx$\"+9dG%R#FI7$Fgx$\"+H9d)[#FI7$F\\y$\"+Vr&Ge#FI7$Fay$\"
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?6&7U7$F.$!+++++$)FE7$FG$!++++e!)FE7$FM$!++++;yFE7$FR$!++++uvFE7$FW$!+
+++KtFE7$Ffn$!++++!4(FE7$F[o$!++++[oFE7$F`o$!++++1mFE7$Feo$!++++kjFE7$
Fjo$!++++AhFE7$F_p$!++++!)eFE7$Fdp$!++++QcFE7$Fip$!++++'R&FE7$F^q$!+++
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urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C
urve 10" "Curve 11" "Curve 12" "Curve 13" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
333 "The blue lines are the perpindicular bisectors and their point of
 intersection the incenter. The red lines are the angle bisectors and \+
their intersection is the circum- center. The green lines are the medi
ans and their intersection is the centroid. And the white lines are th
e altitudes with their intersection being the orthocenter." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Thats a bit too m
uch. Lets just look at the incenter and circumcenter and their circles
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 " incircle( inT, T):       circumcircle( outT, T):\n" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " draw( \{ inT(cr), outT(c
b),  center(inT) (cr), center(outT) (cb), T(fc) \});" }}{PARA 13 "" 1 
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