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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 50 "High School Modul
es > Geometry by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 "    \+
" }{TEXT 256 25 "The Centers of a Triangle" }}{PARA 0 "" 0 "" {TEXT 
-1 113 "\nAn exploration and comparison of four different points which
 can each be considered \"the center\" of a triangle.\n" }}{PARA 0 "" 
0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section fi
rst. Although there will be no output immediately, these definitions a
re used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart; with(plots): with(g
eometry): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1747 "cbk  := 'co
lor = black, filled = true':                      #black\ncwh  := 'col
or = white, filled = true':                      #white\n\ncy   := 'co
lor = COLOR(RGB, .1, .85, .1), filled = true':     #light yellow\ncy1 \+
 := 'color = COLOR(RGB, .8, .7,  .0), filled = true':     #yellow\ncy2
  := 'color = COLOR(RGB, .8, .6,  .0), filled = true':     #yellow (or
angish)\ncc   := 'color = COLOR(RGB, .78, .52, .0), filled = true':   \+
#coral (orange/yellow)\n\ncdb  := 'color = COLOR(RGB, .30, .20, .1), f
illed = true':   #dark brown\ncdb2  := 'color = COLOR(RGB, .45, .30, .
15), filled = true': #dark brown 50% brighter\ncdb3  := 'color = COLOR
(RGB, .6, .40, .2), filled = true':   #dark brown 100% brighter\ncdb4 \+
 := 'color = COLOR(RGB, .75, .50, .25), filled = true': #dark brown 15
0% brighter\ncdb5  := 'color = COLOR(RGB, .85, .70, .5), filled = true
':  #dark brown  brighter\n\ncgr   := 'color = COLOR(RGB, .6, .65, .6)
, filled = true':\ncgr2  := 'color = COLOR(RGB, .35, .38, .35), filled
 = true':\ncgr4  := 'color = COLOR(RGB, .6, .90, .62), filled = true':
  #light green\n\n\n\nco  := 'color = COLOR(RGB, .9, .5, .1), filled =
 true':\nco2 := 'color = COLOR(RGB, .8, .6, .1), filled = true':\ncr  \+
:= 'color = COLOR(RGB, .9, .2, .1), filled = true':\ncr2 := 'color = C
OLOR(RGB, .9, .4, .2), filled = true':\ncy  := 'color = COLOR(RGB, .8,
 .8, .1), filled = true':\n\ncg  := 'color = COLOR(RGB, .35, .6, .35),
 filled = true':\ncg2 := 'color = COLOR(RGB, .5, .75, .5), filled = tr
ue':\ncb  := 'color = COLOR(RGB, .5, .4, .8), filled = true':\ncb2 := \+
'color = COLOR(RGB, .3, .2, .8), filled = true':\ncsb := 'color = COLO
R(RGB, .3,.4, .9), filled = true':\n\ncgr := 'color = COLOR(RGB, .6, .
65, .6), filled = true':\ncgr2 := 'color = COLOR(RGB, .35, .38, .35), \+
filled = true':\n\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 
"lr2 := 'color = red, thickness = 2':\nlb2 := 'color = COLOR(RGB, .3,.
4, .9), thickness = 2':\nlg2 := 'color = COLOR(RGB, .35, .7, .35), thi
ckness = 2':\nlw2 := 'color = white,  thickness = 2':\nlk2 := 'color =
 black,  linestyle = 3':  \nldb2   := 'color = COLOR(RGB, .30, .20, .1
), thickness = 2':\nldb42  := 'color = COLOR(RGB, .85, .70, .5), thick
ness = 2':\nly     := 'color = COLOR(RGB, .8, .6, .0), thickness = 2, \+
linestyle = 3':  \nlgy    := 'color = COLOR(RGB, .4, .4, .4), thicknes
s = 2':  \nlgy2   := 'color = COLOR(RGB, .65, .65, .65), thickness = 2
':  \n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "an := 'axes =
 none':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "point( A, [-3, -
 2]):  \npoint( B, [ 8, - 1]): \npoint( C, [ 0,   5]):\ntriangle( T, [
A,B,C]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "point( A2, [ \+
 0, 0]):  \npoint( B2, [ 10, 0]): \npoint( C2, [  0, 8]):\ntriangle( T
2, [A2,B2,C2]):\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "poin
t( A3, [  0, 0]):  \npoint( B3, [ 10, 0]): \npoint( C3, [  -5, 8]):\nt
riangle( T3, [A3,B3,C3]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
99 "point( A4, [-5,   1]):  \npoint( B4, [ 9, - 1]): \npoint( C4, [ 0,
   5]):\ntriangle( T4, [A4,B4,C4]):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "#draw( [T(cdb),T2(cdb5),T3(cdb3),T4(cdb2)] ); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 46 "1. An
gle Bisectors                >   Incenter" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "A triangle has thr
ee angles. Each angle can be bisected by a line. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "bisector(bi1
,A,T):  bisector(bi2,B,T):  bisector(bi3,C,T):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "draw( [T(cdb), bi1(ldb42)]);" }}}{PARA 0 "" 0 
"" {TEXT -1 138 "\nAmazing these three lines interesect. The intersect
ion of these three lines is the incenter, which is the center of the i
nscribed circle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "draw( [T
(cdb), bi1(ldb42),bi2(ldb42),bi3(ldb42)]);" }}}{PARA 0 "" 0 "" {TEXT 
-1 451 "\n\nEven more amazing is that it is always possible to draw a \+
circle which is centered at this point, which fits just inside the tri
angle so that it barely touches each of the three sides. This is calle
d the \"inscribed circle\" for the triangle. And this is why we call t
his point the \"in center\" because it's the center of the inscribed c
ircle.  Here is a view with the angle bisectors demonstrating they int
ersect at the center of the inscribed circle." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "incircle( in
T, T): \ndraw( [inT(cdb3), T(cdb), bi1(ldb42),bi2(ldb42),bi3(ldb42), c
enter(inT)],an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "draw( [
 inT, center(inT),bi1,bi2,bi3,T], color=[red,black,blue,coral,coral,co
ral], an );" }}}{PARA 0 "" 0 "" {TEXT -1 28 "\n\nHere are other triang
les\n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "draw( T3(cdb),an);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "bisector(bi1,A3,T3):  \+
bisector(bi2,B3,T3):  bisector(bi3,C3,T3): incircle( inT, T3): \ndraw(
 [inT(cdb3), T3(cdb), bi1(ldb42),bi2(ldb42),bi3(ldb42), center(inT)], \+
an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "bisector(bi1,A2,T2
):  bisector(bi2,B2,T2):  bisector(bi3,C2,T2): incircle( inT, T2): \nd
raw( [inT(cdb3), T2(cdb), bi1(ldb42),bi2(ldb42),bi3(ldb42), center(inT
)],an);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 1 " " }{TEXT 262 45 "2. Perpendicular Bisectors   >   Circu
mcenter" }}{PARA 0 "" 0 "" {TEXT -1 3 "\nA " }{TEXT 266 22 "perpendicu
lar bisector" }{TEXT -1 175 " of a segment is another line which bisec
ts the segment and at the same time is perpendicular to it. Each side \+
of a triangle is a line segment, and therefore can be bisected. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
145 "PerpenBisector( p1, A, B):  PerpenBisector( p2, B, C):  PerpenBis
ector( p3, A, C):\nsegment( base, A, B):   draw( [T(cb), p1(ly), base(
lr2) ],an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "segment( bas
e, B, C):   draw( [T(cb), p2(ly), base(lr2) ],an);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 62 "segment( base, A, C):   draw( [T(cb), p3(ly
), base(lr2) ],an);" }}}{PARA 0 "" 0 "" {TEXT -1 109 "\nWhen the perpe
ndicular bisectors are drawn for all three sides, they intersect in a \+
single point called the " }}{PARA 0 "" 0 "" {TEXT 267 12 "circumcenter
" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "draw( \+
[p1(ly),p2(ly),p3(ly),T(cb)],an);" }}}{PARA 0 "" 0 "" {TEXT -1 196 "\n
The circumcenter happens to be the ccenter of the circle which circums
cribes the triangle. Here is a view with the angle bisectors demonstra
ting they intersect the center of the inscribed circle." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "circumc
ircle( outT, T):\ndraw( [ center(outT), p1(ly),p2(ly),p3(ly),T(cb),out
T(cb2)],an);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 120 "Here are other examples. Notice that the circumcenter is
 inside of the triangle above, but not so in this next example.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "PerpenBisector( p1, A3, B3)
:  PerpenBisector( p2, B3, C3):  PerpenBisector( p3, A3, C3):\ncircumc
ircle( outT, T3):\ndraw( [ center(outT), p1(ly),p2(ly),p3(ly),T3(cb),o
utT(cb2)],an);" }}}{PARA 0 "" 0 "" {TEXT -1 127 "\nA right triangle ha
s its circumcenter on the hypotenuse - consistent with a property of s
emi-circles and inscribed triangles.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 179 "PerpenBisector( p1, A2, B2):  PerpenBisector( p2, B2
, C2):  PerpenBisector( p3, A2, C2):\ncircumcircle( outT, T2):\ndraw( \+
[ center(outT), p1(ly),p2(ly),p3(ly),T2(cb),outT(cb2)],an);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "PerpenBisector( p1, A4, B4)
:  PerpenBisector( p2, B4, C4):  PerpenBisector( p3, A4, C4):\ncircumc
ircle( outT, T4):\ndraw( [ center(outT), p1(ly),p2(ly),p3(ly),T4(cb),o
utT(cb2)],an);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 1 " " }{TEXT 268 53 "3. Comparing Inscribed Circle \+
to Circumscribed Circle" }}{PARA 0 "" 0 "" {TEXT -1 207 "\nWe have so \+
far seen that each triangle has an inscribed circle and a circumscribe
d circle. What does it look like if we plot both at once? Do they have
 the same center? Lets look at both on the same graph.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "bisector(bi1,A,T):  bisector(bi2,B
,T):  bisector(bi3,C,T):  incircle( inT,T):\nPerpenBisector( p1,A,B): \+
 PerpenBisector( p2,B,C):  PerpenBisector( p3,A,C):\ncircumcircle( out
T, T):\ndraw( [center(inT),center(outT), inT(cb),T(cwh),\n        bi1(
lw2),bi2(lw2),bi3(lw2),p1(ly),p2(ly),p3(ly), outT(cb2)],an);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "draw( [center(inT),center(ou
tT), inT(cb),T(cwh), outT(cb2)],an);" }}}{PARA 0 "" 0 "" {TEXT -1 168 
" \nThe small circle is the inscribed circle and the large one is the \+
circumscribed circle. You can also see that the incenter and circumcen
ters are not the same at all.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 316 "bisector(bi1,A3,T3):  bisector(bi2,B3,T3):  bisector(bi3,C3,T
3):  incircle( inT,T3):\nPerpenBisector( p1,A3,B3):  PerpenBisector( p
2,B3,C3):  PerpenBisector( p3,A3,C3):\ncircumcircle( outT, T3):\ndraw(
 [center(inT),center(outT), inT(cb),T3(cwh),\n        bi1(lw2),bi2(lw2
),bi3(lw2),p1(ly),p2(ly),p3(ly), outT(cb2)],an);\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 64 "draw( [center(inT),center(outT), inT(cb),T(
cwh), outT(cb2)],an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "i
ncircle( inT,T2):      circumcircle( outT, T2):\ndraw( [center(inT),ce
nter(outT), inT(cb),T2(cwh), outT(cb2)],an);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 114 "incircle( inT,T4):      circumcircle( outT, T4)
:\ndraw( [center(inT),center(outT), inT(cb),T4(cwh), outT(cb2)],an);" 
}}}{PARA 0 "" 0 "" {TEXT -1 287 "\n\n\nIf we turn this situation aroun
d and start with two circles, can you find a unique triangle which has
 the smaller circle as the inscribed circle, and larger as the circums
cribed circle? What if you are also given a single point on the larger
 circle which is a vertex of the triangle.\n" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 104 "incircle( inT,T2):      \ncircumcircle( outT, T):
\ndraw( [inT(color=navy),outT(color=navy) ],axes = none);" }}}{PARA 0 
"" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }
{TEXT 263 48 "4. Medians                          >   Centroid" }}
{PARA 0 "" 0 "" {TEXT -1 204 "\nA median of a triangle is a line which
 connects a vertex with the midpoint of the opposite side. Note that a
 median is usually not an angle bisector or a perpendicular bisector, \+
except in special cases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "median(md1,A,T):  median(md2,B,T):
  median(md3,C,T):\nmidpoint(mp,B,C): segment( s1,B,mp):segment( s2,C,
mp):\ndraw( [T(co), md1(lk2), mp, s1(lb2), s2(lg2)], an);" }}}{PARA 0 
"" 0 "" {TEXT -1 264 "\nIf you look closely, you'll see that the black
 line is drawn from the vertex to the midpoint of the opposite side. T
he opposite side is cut into two pieces, one green and one blue, and t
hese pieces are of equal length. Here is another median of the same tr
iangle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "midpoint(mp,A,C)
: segment( s1,A,mp):segment( s2,C,mp):\ndraw( [T(co), md2(lk2), mp, s1
(lb2), s2(lg2)], an);" }}}{PARA 0 "" 0 "" {TEXT -1 55 "\n\nThe point w
here the three medians meet is called the " }{TEXT 269 8 "centroid" }
{TEXT -1 262 ". This is a very special point because it represents the
 center of gravity of the triangle. If the triangle were made out of m
etal or wood of even thickness, the centroid would be the only point w
here you could balance the triangle on the tip of a single finger!" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "draw( [T(co), md1(lk2),md2(l
k2),md3(lk2)],an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "medi
an(md1,A3,T3):  median(md2,B3,T3):  median(md3,C3,T3):\ndraw( [T3(co),
 md1(lk2),md2(lk2),md3(lk2)],an);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "median(md1,A2,T2):  median(md2,B2,T2):  median(md3,C
2,T2):\ndraw( [T2(co), md1(lk2),md2(lk2),md3(lk2)],an);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "median(md1,A4,T4):  median(md2,B4,
T4):  median(md3,C4,T4):\ndraw( [T4(co), md1(lk2),md2(lk2),md3(lk2)],a
n);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "
The centroid has another special property. It cuts each median into tw
o pieces, one twice as long as the other. We can verify this. Note tha
t the distance from the centroid to each vertex is exactly twice the d
istance from the centroid to midpoint." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "centroid( ct, T):  midpo
int( mAB, A, B):  midpoint( mBC, B, C):  midpoint( mAC, A, C):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "evalf( distance( C, ct)) ; \+
  \011evalf( distance( mAB, ct)) ; \n`Twice the second piece is `= 2*%
; `Which is the same as the first piece`;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 136 "evalf( distance( A, ct)) ;    \011evalf( distance(
 mBC, ct)) ; \n`Twice the second piece is `= 2*%; `Which is the same a
s the first piece`; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "ev
alf( distance( B, ct)) ; \011\011evalf( distance( mAC, ct)) ; \n`Twice
 the second piece is `= 2*%; `Which is the same as the first piece`; \+
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " 
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 52 "5. Altitudes \+
                        >   Orthocenter" }}{PARA 0 "" 0 "" {TEXT -1 1 
"\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 372 "A
n altitude is a line segment that extends from one vertex to the oppos
ite side, or an extension of that opposite side, and is perpendicular \+
to it. An altitude is somewhat like a median in that it goes from vert
ex to opposite side, however, it does not necessarily pass through the
 midpoint of the opposite side. Many times, the altitude is located ou
tside of the circle. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 93 "altitude(alt1,A,T):  altitude(alt2,B,T): \+
 altitude(alt3,C,T):\ndraw( [T(cg), alt3(lgy2)],an );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "draw( [T(cg), alt1(lgy2)],an );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "draw( [T(cg), alt2(lgy2)],a
n );" }}}{PARA 0 "" 0 "" {TEXT -1 56 "\n\nThe intersection of the thre
e altitudes is called the " }{TEXT 270 11 "orthocenter" }{TEXT -1 2 ".
\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "draw( [T(cg), alt1(lgy
2),alt2(lgy2), alt3(lgy2)],an );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 85 "Lets see some more examples. Some of th
e orthocenters are not in the triangle itself." }}{PARA 0 "" 0 "" 
{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "altitude(
alt1,A3,T3):  altitude(alt2,B3,T3):  altitude(alt3,C3,T3):\northocente
r( oc, T3):\ndraw( [oc, T3(cg), alt1(lgy2),alt2(lgy2), alt3(lgy2)],an \+
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "altitude(alt1,A2,T2)
:  altitude(alt2,B2,T2):  altitude(alt3,C2,T2):\northocenter( oc, T2):
\ndraw( [oc, T2(cg), alt1(lgy2),alt2(lgy2), alt3(lgy2)],an );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "altitude(alt1,A4,T4):  alti
tude(alt2,B4,T4):  altitude(alt3,C4,T4):\northocenter( oc, T4):\ndraw(
 [oc, T4(cg), alt1(lgy2),alt2(lgy2), alt3(lgy2)],an );" }}}{PARA 0 "" 
0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }
{TEXT 265 41 "6. All Together Now  - Compare & Contrast" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We've seen all four
 points of concurrency. How do they look together? Lets draw all 12 of
 the lines discussed on a single wild diagram." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "bisector(bi
1,A,T):  bisector(bi2,B,T):  bisector(bi3,C,T): incircle( inT, T): \nP
erpenBisector( p1,A,B):  PerpenBisector( p2,B,C): PerpenBisector( p3,A
,C):\ncircumcircle( outT, T):centroid( ct, T):orthocenter( oc, T):\nme
dian(md1,A,T):  median(md2,B,T):  median(md3,C,T):\naltitude(alt1,A,T)
:  altitude(alt2,B,T):  altitude(alt3,C,T):\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 218 "draw( [  center(inT),center(outT),oc,ct,\n     \+
    bi1,bi2,bi3,                   p1(ly),p2(ly),p3(ly),\n         md1
(lk2),md2(lk2),md3(lk2),    alt1(lgy2),alt2(lgy2),alt3(lgy2),\n       \+
  inT(cb),T(cwh),outT(cb2) ],an );" }}}{PARA 0 "" 0 "" {TEXT -1 62 "\n
\nWell thats' an eyefull! Lets just look at the four centers.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "draw( [  center(inT),center(
outT),oc,\nct,         inT(cb),T(cwh),outT(cb2) ],an );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "draw( [  center(inT),center(outT),o
c,ct,inT(cy2),T(co)],an );" }}}{PARA 0 "" 0 "" {TEXT -1 59 "\nAs you c
an see, the four centers are four distinct points!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 145 "incircle( inT, T3):circumcircle( outT, T3):centro
id( ct, T3):orthocenter( oc, T3):\ndraw( [  center(inT),center(outT),o
c,ct,inT(cy2),T3(co)], an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
145 "incircle( inT, T2):circumcircle( outT, T2):centroid( ct, T2):orth
ocenter( oc, T2):\ndraw( [  center(inT),center(outT),oc,ct,inT(cy2),T2
(co)], an);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "incircle( i
nT, T4):circumcircle( outT, T4):centroid( ct, T4):orthocenter( oc, T4)
:\ndraw( [  center(inT),center(outT),oc,ct,inT(cy2),T4(co)], an);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n         \251 2002 Water
loo Maple Inc " }}}{MARK "0 1" 31 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }