##+##########################################################################
#
# Triangle Madness -- Shows some interesting properties of a triangle
# by Keith Vetter - July 7, 2003
# January 10, 2007 -- added 9 point circle
# February 10, 2021 -- added more Morley's Miracle triangles
#

package require Tk
catch {package require tile}
package require textutil
array set P {1 {17 -227} 2 {-215 66} 3 {109 119}} ;# Initial position
set S(title) "Triangle Madness"

set S(circumcenter) 1
set S(incenter) 1
set S(centroid) 0
set S(orthocenter) 0
set S(eulerline) 0
set S(ninepointcircle) 0
set S(morley'smiracle) 0
set S(morley'smiracle2) 0
set S(ninepointcircle) 0

set C(circumcenter) magenta
set C(incenter) deeppink
set C(centroid) navy
set C(orthocenter) DeepSkyBlue2
set C(eulerline) black
set C(9point) blueviolet

proc DoDisplay {} {
    global P S

    wm title . $S(title)
    pack [frame .ctrl -relief ridge -bd 2 -padx 5 -pady 5] \
        -side right -fill both -ipady 5

    button .about -text About -command About
    .about configure  -font "[font actual [.about cget -font]] -weight bold"
    option add *font [.about cget -font]
    option add *Checkbutton.anchor w
    option add *Checkbutton.relief raised
    button .all_help -text Help -command {Help "" all}

    foreach txt [list Circumcenter Incenter Centroid Orthocenter \
                     "Euler Line" "Nine Point Circle" "Morley's Miracle" "Morley's Miracle 2"] {
        set w [string tolower [string map {" " ""} $txt]]
        checkbutton .$w -variable S($w) -text $txt -command DrawLines
        button .h_$w -text "?" -command [list Help $txt $w]
        grid .$w .h_$w -in .ctrl -sticky news
    }

    grid rowconfigure .ctrl 50 -weight 10
    grid .all_help - -in .ctrl -row 100 -sticky ew
    grid .about - -in .ctrl -row 101 -sticky ew

    canvas .c -width 1000 -height 1000 -bd 2 -relief raised
    bind all <Alt-c> {console show}
    bind .c <Configure> {ReCenter %W %h %w}
    pack .c -side top -fill both -expand 1

    foreach w {1 2 3} {
        .c create oval [Box $P($w)] -tag [list vert p$w] -fill red
        .c bind p$w <B1-Motion> [list DoButton $w %x %y]
    }
    bind all <Alt-c> [list console show]
}
##+##########################################################################
#
# Recenter -- keeps 0,0 at the center of the canvas during resizing
#
proc ReCenter {W h w} {                   ;# Called by configure event
    set h2 [expr {$h / 2}]
    set w2 [expr {$w / 2}]
    $W config -scrollregion [list -$w2 -$h2 $w2 $h2]
}
##+##########################################################################
#
# DoButton -- interactively moves a vertex around and redraws everything
#
proc DoButton {who X Y} {
    foreach {x y} $::P($who) {X Y} [list [.c canvasx $X] [.c canvasy $Y]] break
    set ::P($who) [list $X $Y]
    .c move p$who [expr {$X - $x}] [expr {$Y - $y}]
    DrawLines
}
##+##########################################################################
#
# Box -- returns coordinates for a box around a given point
#
proc Box {xy {r 5}} {
    foreach {x y} $xy break
    return [list [expr {$x-$r}] [expr {$y-$r}] [expr {$x+$r}] [expr {$y+$r}]]
}
##+##########################################################################
#
# VAdd -- adds two vectors w/ scaling of 2nd vector
# VSub -- subtract two vectors
# VNormal -- returns normal vector to v
#
proc VAdd {v1 v2 {scaling 1}} {
    foreach {x1 y1} $v1 {x2 y2} $v2 break
    return [list [expr {$x1 + $scaling*$x2}] [expr {$y1 + $scaling*$y2}]]
}
proc VSub {v1 v2} { return [VAdd $v1 $v2 -1] }
proc VNormal {v} { foreach {x y} $v break; return [list $y [expr {-1 * $x}]]}
proc VReflect {p1 reflect_pt} {
    # reflects point p1 about $reflect_pt so new point is collinear and equally spaced
    return [VAdd $reflect_pt [VSub $reflect_pt $p1]]
}
##+##########################################################################
#
# DrawLines -- draws all the various lines on the screen
#
proc DrawLines {} {
    global P
    .c delete outer circum incenter centroid ortho euler equi morley morley2

    .c create poly [concat $P(1) $P(2) $P(3)] -tag outer -width 2 \
        -fill lightgreen -outline black
    set theta [FindAngle3 $P(1) $P(2) $P(3)]    ;# Check for collinear points
    if {$theta != 0} {
        CircumCenter
        InCenter
        Centroid
        OrthoCenter
        EulerLine
        9PointCircle
        Morley
        Morley2
    }
    .c raise vert
}
##+##########################################################################
#
# IntersectV -- find where 2 point/vector intersect
#
# p1+K(v1) = p3+J(v3)
# convert into and solve matrix equation (a b / c d) ( K / J) = ( e / f )
#
proc IntersectV {p1 v1 p3 v3} {
    foreach {x1 y1} $p1 {vx1 vy1} $v1 {x3 y3} $p3 {vx3 vy3} $v3 break

    set a $vx1
    set b [expr {-1 * $vx3}]
    set c $vy1
    set d [expr {-1 * $vy3}]
    set e [expr {$x3 - $x1}]
    set f [expr {$y3 - $y1}]

    set det [expr {double($a*$d - $b*$c)}]
    if {$det == 0} {error "Determinant is 0"}

    set k [expr {($d*$e - $b*$f) / $det}]
    #set j [expr {($a*$f - $c*$e) / $det}]
    return [VAdd $p1 $v1 $k]
}
##+##########################################################################
#
# Intersect -- find where two line intersect given two points on each line
#
proc Intersect {p1 p2 p3 p4} {
    return [IntersectV $p1 [VSub $p2 $p1] $p3 [VSub $p4 $p3]]
}
##+##########################################################################
#
# TrisectAngle -- returns two points which are on the two lines trisecting
# the angle created by points p1,p2,p3. We use the cross product to tell
# us clockwise ordering.
#
proc TrisectAngle {p1 p2 p3} {
    set cross [Cross [VSub $p2 $p1] [VSub $p2 $p3]]
    if {$cross < 0} {foreach {p1 p3} [list $p3 $p1] break}

    set theta [FindAngle3 $p1 $p2 $p3]          ;# What the angle is
    set theta1 [expr {$theta / 3.0}]            ;# 1/3 of that angle
    set theta2 [expr {2 * $theta1}]             ;# 2/3 of that angle

    set v [VSub $p3 $p2]                        ;# We'll rotate this leg
    set v1 [RotateCC $v $theta1]                ;# By 1/3
    set v2 [RotateCC $v $theta2]                ;# By 2/3
    set t1 [VAdd $p2 $v1]                       ;# Trisect point 1
    set t2 [VAdd $p2 $v2]                       ;# Trisect point 2

    if {$cross < 0} { foreach {t1 t2} [list $t2 $t1] break }
    return [list $t1 $t2]
}
##+##########################################################################
#
# BisectAngle -- returns point on bisector of an angle
#
proc BisectAngle {p1 p2 p3} {
    foreach {x1 y1} [VSub $p1 $p2] {x2 y2} [VSub $p3 $p2] break
    set s1 [expr {100.0 / hypot($x1, $y1)}]
    set s2 [expr {100.0 / hypot($x2, $y2)}]
    set v1 [VAdd $p2 [list $x1 $y1] $s1]        ;# Unit vector from p2 to p1
    set v2 [VAdd $p2 [list $x2 $y2] $s2]        ;# Unit vector from p2 to p3
    return [VAdd $v1 [VSub $v2 $v1] .5]
}
##+##########################################################################
#
# FindAngle3 -- returns the angle between three points
#
proc FindAngle3 {p1 p2 p3} {
    foreach {x1 y1} [VSub $p1 $p2] {x2 y2} [VSub $p3 $p2] break

    set m1 [expr {hypot($x1,$y1)}]
    set m2 [expr {hypot($x2,$y2)}]
    if {$m1 == 0 || $m2 == 0} { return 0 }      ;# Coincidental points
    set dot [expr {$x1 * $x2 + $y1 * $y2}]

    set theta [expr {acos($dot / $m1 / $m2)}]
    if {$theta < 1e-5} {set theta 0}
    return $theta
}
##+##########################################################################
#
# RotateCC -- rotates vector v by beta radians counter-clockwise
#
proc RotateCC {v beta} {
    foreach {x y} $v break
    set xx [expr {$x * cos(-$beta) - $y * sin(-$beta)}]
    set yy [expr {$x * sin(-$beta) + $y * cos(-$beta)}]
    return [list $xx $yy]
}
##+##########################################################################
#
# Cross -- returns the cross product -- easy w/ z=0
#
proc Cross {v1 v2} {
    foreach {x1 y1} $v1 {x2 y2} $v2 break
    return [expr {($x1*$y2) - ($y1*$x2)}]
}
proc About {} {
    set msg "$::S(title)\nby Keith Vetter, July 2003\n\n"
    append msg "A little program to visualize some of the many interesting "
    append msg "properties of a triangle. You can grab and drag any of the "
    append msg "triangle's vertices. Click on the \"?\" next to a property "
    append msg "to learn more about it."
    #tk_messageBox -title "About $::S(title)" -message "$title$msg"
    MyHelpBox "About $::S(title)" "$msg"
}
proc CircumCenter {} {
    global P S C

    .c delete circum
    set h12 [VAdd $P(1) [VSub $P(2) $P(1)] .5]  ;# Midpoints of each side
    set h13 [VAdd $P(1) [VSub $P(3) $P(1)] .5]
    set h23 [VAdd $P(2) [VSub $P(3) $P(2)] .5]

    set n12 [VNormal [VSub $P(2) $P(1)]]        ;# Normal to side p1-p2
    set n13 [VNormal [VSub $P(3) $P(1)]]
    set O [IntersectV $h12 $n12 $h13 $n13]      ;# The circumcenter
    set S(v,circumcenter) $O
    foreach {rx ry} [VSub $P(1) $O] break       ;# Radius vector
    set radius [expr {hypot($rx,$ry)}]          ;# Radius magnitude

    if {! $S(circumcenter) && ! $S(eulerline)} return
    .c create text $O -text "O" -anchor se -tag circum -fill $C(circumcenter)
    .c create oval [Box $O 3] -tag circum -fill $C(circumcenter) -outline $C(circumcenter)
    if {! $S(circumcenter)} return
    .c create line [concat $h12 $O] -tag circum -fill $C(circumcenter)
    .c create line [concat $h13 $O] -tag circum -fill $C(circumcenter)
    .c create line [concat $h23 $O] -tag circum -fill $C(circumcenter)
    .c create oval [Box $O $radius] -tag circum -outline $C(circumcenter)
    .c create line [concat $O [VAdd $O [list $radius 0]]] -tag circum \
        -fill $C(circumcenter) -dash 1
}
proc InCenter {} {
    global P S C

    .c delete incenter
    set b1 [BisectAngle $P(3) $P(1) $P(2)]      ;# Bisect angle 1
    set b2 [BisectAngle $P(1) $P(2) $P(3)]      ;# Bisect angle 2
    set Q [Intersect $P(1) $b1 $P(2) $b2]
    set S(v,incenter) $Q

    # Need distance from any side to the incenter
    foreach {qx qy p1x p1y p2x p2y} [concat $Q $P(1) $P(2)] break
    set vqx [expr {$qx - $p1x}]
    set vqy [expr {$qy - $p1y}]
    set vpx [expr {$p2x - $p1x}]
    set vpy [expr {$p2y - $p1y}]
    set radius [expr {($vqx * $vpy - $vqy * $vpx) / hypot($vpx, $vpy)}]

    if {! $S(incenter)} return
    .c create text $Q -text "Q" -anchor se -tag incenter -fill $C(incenter)
    .c create oval [Box $Q 3] -tag incenter -fill $C(incenter) -outline $C(incenter)
    .c create line [concat $P(1) $Q] -tag incenter -fill $C(incenter)
    .c create line [concat $P(2) $Q]  -tag incenter -fill $C(incenter)
    .c create line [concat $P(3) $Q]  -tag incenter -fill $C(incenter)
    .c create oval [Box $Q $radius] -tag incenter -outline $C(incenter)
    .c create line [concat $Q [expr {$qx + $radius}] $qy] -tag incenter \
        -fill $C(incenter) -dash 1
}
proc Centroid {} {
    global P S C

    .c delete centroid
    set h12 [VAdd $P(1) [VSub $P(2) $P(1)] .5]  ;# Midpoints of each side
    set h13 [VAdd $P(1) [VSub $P(3) $P(1)] .5]
    set h23 [VAdd $P(2) [VSub $P(3) $P(2)] .5]
    set CC [Intersect $P(1) $h23 $P(2) $h13]
    set S(v,centroid) $CC

    if {! $S(centroid) && ! $S(eulerline)} return
    set col $C(centroid)
    .c create text $CC -text "C" -anchor se -tag centroid -fill $col
    .c create oval [Box $CC 3] -tag centroid -fill $col -outline $col
    if {! $S(centroid)} return
    .c create line [concat $P(1) $h23] -tag centroid -fill $col
    .c create line [concat $P(2) $h13] -tag centroid -fill $col
    .c create line [concat $P(3) $h12] -tag centroid -fill $col
}
proc OrthoCenter {} {
    global P S C

    .c delete ortho
    set v1 [VSub $P(3) $P(2)]                   ;# Vector for side p2-p3
    set a1 [IntersectV $P(1) [VNormal $v1] $P(2) $v1]
    set v2 [VSub $P(3) $P(1)]
    set a2 [IntersectV $P(2) [VNormal $v2] $P(1) $v2]
    set v3 [VSub $P(2) $P(1)]
    set a3 [IntersectV $P(3) [VNormal $v3] $P(1) $v3]
    set H [Intersect $P(1) $a1 $P(2) $a2]
    set S(v,orthocenter) $H

    if {! $S(orthocenter) && ! $S(eulerline)} return
    set col $C(orthocenter)
    .c create text $H -text "H" -anchor se -tag ortho -fill $col
    .c create oval [Box $H 3] -tag ortho -fill $col -outline $col
    if {! $S(orthocenter)} return
    .c create line [concat $P(1) $a1] -tag ortho -fill $col
    .c create line [concat $P(2) $a2] -tag ortho -fill $col
    .c create line [concat $P(3) $a3] -tag ortho -fill $col

    # Altitude's feet maybe outside triangle, draw a dashed line to look bettter
    .c create line [concat $P(2) $P(3) $a1] -tag {x ortho} -fill $col -dash 1
    .c create line [concat $P(1) $P(3) $a2] -tag {x ortho} -fill $col -dash 1
    .c create line [concat $P(1) $P(2) $a3] -tag {x ortho} -fill $col -dash 1
    .c lower x
}
proc EulerLine {} {
    global P S C
    if {! $S(eulerline)} return

    .c delete euler
    .c create line [concat $S(v,orthocenter) $S(v,centroid)] -tag euler
    .c create line [concat $S(v,circumcenter) $S(v,centroid)] -tag euler
    .c itemconfig euler -fill $C(eulerline) -width 3
}
proc 9PointCircle {} {
    global P S C

    .c delete 9point
    if {! $S(ninepointcircle) && ! $S(eulerline)} return

    # 3 side medians
    set h12 [VAdd $P(1) [VSub $P(2) $P(1)] .5]  ;# Midpoints of each side
    set h13 [VAdd $P(1) [VSub $P(3) $P(1)] .5]
    set h23 [VAdd $P(2) [VSub $P(3) $P(2)] .5]

    # 3 feet of altitudes
    set v1 [VSub $P(3) $P(2)]                   ;# Vector for side p2-p3
    set a1 [IntersectV $P(1) [VNormal $v1] $P(2) $v1]
    set v2 [VSub $P(3) $P(1)]
    set a2 [IntersectV $P(2) [VNormal $v2] $P(1) $v2]
    set v3 [VSub $P(2) $P(1)]
    set a3 [IntersectV $P(3) [VNormal $v3] $P(1) $v3]
    set H [Intersect $P(1) $a1 $P(2) $a2]       ;# Orthocenter

    # 3 midpoints of orthocenter to vertices
    set o1 [VAdd $P(1) [VSub $H $P(1)] .5]
    set o2 [VAdd $P(2) [VSub $H $P(2)] .5]
    set o3 [VAdd $P(3) [VSub $H $P(3)] .5]

    foreach {O radius} [GetCenter $h12 $h13 $h23] break
    set xy [Box $O 3]
    .c create oval $xy -tag 9point -fill $C(9point) -outline $C(9point)
    .c create text $O -text "9" -anchor se -tag 9point -fill $C(9point)
    if {! $S(ninepointcircle)} return

    foreach who {h12 h13 h23 a1 a2 a3 o1 o2 o3 O} {
        set xy [Box [set $who] 3]
        .c create oval $xy -tag 9point -fill $C(9point) -outline $C(9point)
    }
    set xy [Box $O $radius]
    .c create oval $xy -tag 9point -fill {} -outline $C(9point)
    .c create line [concat $O [VAdd $O [list $radius 0]]] -tag 9point \
        -fill $C(9point) -dash 1
}
proc GetCenter {p1 p2 p3} {
    set h12 [VAdd $p1 [VSub $p2 $p1] .5]        ;# Midpoints of each side
    set h13 [VAdd $p1 [VSub $p3 $p1] .5]
    set h23 [VAdd $p2 [VSub $p3 $p2] .5]

    set n12 [VNormal [VSub $p2 $p1]]            ;# Normal to side p1-p2
    set n13 [VNormal [VSub $p3 $p1]]
    set O [IntersectV $h12 $n12 $h13 $n13]      ;# The circumcenter
    set S(v,circumcenter) $O
    foreach {rx ry} [VSub $p1 $O] break         ;# Radius vector
    set radius [expr {hypot($rx,$ry)}]          ;# Radius magnitude
    return [list $O $radius]
}
##+##########################################################################
#
# Morley -- draws the angle trisectors out to where they
# meet and then draws the Morley triangle in the middle.
#
proc Morley {} {
    global P

    .c delete morley
    if {! $::S(morley'smiracle) && ! $::S(morley'smiracle2)} return

    # Get trisector lines out of each vertex
    foreach {t(1,1) t(1,2)} [TrisectAngle $P(3) $P(1) $P(2)] break
    foreach {t(2,1) t(2,2)} [TrisectAngle $P(1) $P(2) $P(3)] break
    foreach {t(3,1) t(3,2)} [TrisectAngle $P(2) $P(3) $P(1)] break

    # Find where trisector line segments intersect
    set E1 [Intersect $P(1) $t(1,1) $P(2) $t(2,2)]
    set E2 [Intersect $P(2) $t(2,1) $P(3) $t(3,2)]
    set E3 [Intersect $P(1) $t(1,2) $P(3) $t(3,1)]
    if {$E1 == {} || $E2 == {} || $E3 == {}} return ;# Colinear lines?

    .c create line [concat $P(1) $E1 $P(2) $E2 $P(3) $E3 $P(1)] -tag morley \
        -fill blue
    .c create line [concat $E1 $E2 $E3 $E1] -tag morley -fill red -width 2
}
##+##########################################################################
#
# Morley -- 4 equilateral triangles formed from intersecting
# trisectors of external angles
#
proc Morley2 {} {
    global P

    .c delete morley2
    if {! $::S(morley'smiracle2)} return

    # For exterior angles we need reflected points
    foreach p {1 2 3} {
        foreach reflect {1 2 3} {
            set P($p,$reflect) [VReflect $P($p) $P($reflect)]
        }
    }

    # Get 6 trisector lines out of each vertex
    lassign [TrisectAngle $P(3) $P(1) $P(2,1)] tr(1,1) tr(1,2)
    lassign [TrisectAngle $P(2) $P(1) $P(3)] tr(1,3) tr(1,4)
    lassign [TrisectAngle $P(3,1) $P(1) $P(2)] tr(1,5) tr(1,6)
    lassign [TrisectAngle $P(1) $P(2) $P(3,2)] tr(2,1) tr(2,2)
    lassign [TrisectAngle $P(3) $P(2) $P(1)] tr(2,3) tr(2,4)
    lassign [TrisectAngle $P(1,2) $P(2) $P(3)] tr(2,5) tr(2,6)
    lassign [TrisectAngle $P(2) $P(3) $P(1,3)] tr(3,1) tr(3,2)
    lassign [TrisectAngle $P(1) $P(3) $P(2)] tr(3,3) tr(3,4)
    lassign [TrisectAngle $P(2,3) $P(3) $P(1)] tr(3,5) tr(3,6)

    # Find where trisector line segments intersect
    set E1 [Intersect $P(1) $tr(1,5) $P(2) $tr(2,2)]
    set E2 [Intersect $P(1) $tr(1,6) $P(3) $tr(3,4)]
    set E3 [Intersect $P(2) $tr(2,1) $P(3) $tr(3,3)]

    set E4 [Intersect $P(2) $tr(2,5) $P(3) $tr(3,2)]
    set E5 [Intersect $P(2) $tr(2,6) $P(1) $tr(1,4)]
    set E6 [Intersect $P(1) $tr(1,3) $P(3) $tr(3,1)]

    set E7 [Intersect $P(3) $tr(3,5) $P(1) $tr(1,2)]
    set E8 [Intersect $P(3) $tr(3,6) $P(2) $tr(2,4)]
    set E9 [Intersect $P(1) $tr(1,1) $P(2) $tr(2,3)]
    if {$E1 == {} || $E2 == {} || $E3 == {}} return ;# Colinear lines?
    if {$E4 == {} || $E5 == {} || $E6 == {}} return ;# Colinear lines?
    if {$E7 == {} || $E8 == {} || $E9 == {}} return ;# Colinear lines?

    set xy [concat $P(1) $E1 $P(2) $E3 $P(3) $E2 $P(1)]
    .c create line $xy -tag morley2 -fill blue
    .c create line [concat $E1 $E2 $E3 $E1] -tag morley2 -fill red -width 4

    set xy [concat $P(2) $E4 $P(3) $E6 $P(1) $E5 $P(2)]
    .c create line $xy -tag morley2 -fill blue
    .c create line [concat $E4 $E5 $E6 $E4] -tag morley2 -fill red -width 4

    set xy [concat $P(3) $E7 $P(1) $E9 $P(2) $E8 $P(3)]
    .c create line $xy -tag morley2 -fill blue
    .c create line [concat $E7 $E8 $E9 $E7] -tag morley2 -fill red -width 4

    .c create line [concat $E1 $E4 $E7 $E1] -tag morley2 -fill red -width 4

}
##+##########################################################################
#
# MyHelpBox -- like tk_messageBox but w/o the grab
#
proc MyHelpBox {title msg} {
    set W .help

    set msg [string trim $msg]
    if {[winfo exists $W]} {
        wm title $W $title
        $W.msg config -text $msg
        $W.msg config -wraplength [expr {[string length $msg] > 1000 ? "5i" : "3i"}]
        return
    }

    destroy $W
    toplevel $W
    wm transient $W .
    wm title $W $title
    wm resizable $W 0 0
    set bg [$W cget -background]
    frame $W.bot -background $bg
    pack $W.bot -side bottom -fill both
    frame $W.top -background $bg
    pack $W.top -side top -fill both -expand 1
    set windowingsystem [tk windowingsystem]
    if {$windowingsystem ne "classic" && $windowingsystem ne "aqua"} {
        $W.bot configure -relief raised -bd 1
        $W.top configure -relief raised -bd 1
    }
    label $W.icon -image ::img::info -bg $bg
    label $W.msg -anchor nw -justify left -text $msg -bg $bg -wraplength 3i
    $W.msg config -wraplength [expr {[string length $msg] > 1000 ? "5i" : "3i"}]
    grid $W.icon $W.msg -in $W.top -sticky news -padx 2m -pady 2m
    grid config $W.icon -sticky n
    grid columnconfigure $W.top 1 -weight 1
    grid rowconfigure $W.top 0 -weight 1

    if {[info commands ::ttk::button] ne {}} {
        ::ttk::button $W.ok -text "OK" -command [list destroy $W]
        $W.top config -bd 0
        $W.bot config -bd 0
    } else {
        button $W.ok -text "OK" -padx 3m -command [list destroy $W]
    }
    pack $W.ok -in $W.bot -padx 3m -pady 2m -expand 1
    bind $W <space> [list destroy $W]
    ::tk::PlaceWindow $W widget .
}
image create photo ::img::info -data {
    R0lGODlhIAAgALMAAAAAAAAA/4SEhMbGxvf/Mf//////////////////////////////////////
    /////yH5BAEAAAQALAAAAAAgACAAAAStkMhJibj41s0nHkUoDljXXaCoqqRgUkK6zqP7CvQQ7IGs
    AiYcjcejFYAb4ZAYMB4rMaeO51sNkBKlc/uzRbng0NWlnTF3XAAZzExj2ET3BV7cqufctv2Tj0vv
    Fn11RndkVSt6OYVZRmeDXRoTAGFOhTaSlDOWHACHW2MlHQCdYFebN6OkVqkZlzcXqTKWoS8wGJMh
    s7WoIoC7v7i+v7uTwsO1o5HHu7TLtcodEQAAOw==}

proc Help {title who} {
    global m
    set m(circumcenter) "The circumcenter is the center of a circle that "
    append m(circumcenter) "passes through each vertex of a triangle. It is "
    append m(circumcenter) "the intersection the lines perpendicular to and "
    append m(circumcenter) "passing through the midpoint of each side."
    set m(incenter) "The incenter is the center of a circle that is tangent "
    append m(incenter) "to each side of the triangle. It is the intersection "
    append m(incenter) "of the lines bisecting each vertex."
    set m(centroid) "The centroid is the balance point of the triangle. "
    append m(centroid) "It is the intersection of the lines from each vertex "
    append m(centroid) "to the opposite median."
    set m(orthocenter) "The orthocenter is the intersection of the "
    append m(orthocenter) "three altitudes of a triangle."
    set m(eulerline) "Euler Line is the lined formed by the collinear points: "
    append m(eulerline) "centroid, circumcenter, othrocenter and the center "
    append m(eulerline) "of the nine-point circle."
    set m(ninepointcircle) "The nine-point circle is also known as both the "
    append m(ninepointcircle) "Euler Circle and Feuerbach Circle and is a "
    append m(ninepointcircle) "circle passing through nine seemingly random "
    append m(ninepointcircle) "points: the midpoints of the three sides, the "
    append m(ninepointcircle) "feet of the three altitudes, and the midpoints "
    append m(ninepointcircle) "of the three line segments connecting the "
    append m(ninepointcircle) "orthocenter to the vertices."
    set m(morley'smiracle) "Morley's Miracle is an equilateral triangle formed "
    append m(morley'smiracle) "by the three points of intersection of the "
    append m(morley'smiracle) "adjacent trisectors of the angles of a triangle."
    set m(morley'smiracle2) "Morley's Miracle can be extended by considering "
    append m(morley'smiracle2) "the intersection of the trisectors of the "
    append m(morley'smiracle2) "exterior angles."

    if {![info exists m(all)]} {
        set m(all) ""
        foreach txt [list Circumcenter Incenter Centroid Orthocenter \
                         "Euler Line" "Nine Point Circle" "Morley's Miracle" \
                         "Morley's Miracle 2"] {
            set w [string tolower [string map {" " ""} $txt]]
            append m(all) $txt \n\n $m($w) \n\n
        }
    }
    set msg "$title\n\n$m($who)"

    #tk_messageBox -title "$::S(title) -- $title" -message $msg
    MyHelpBox "$::S(title) -- $title" $msg
}
################################################################
DoDisplay
DrawLines
return