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<html>
<head>
<title>JSXGraph example</title>
<link rel="stylesheet" type="text/css" href="../distrib/jsxgraph.css" />
<script type="text/javascript" src="/javascript/prototype/prototype.js"></script>
<script type="text/javascript" src="../distrib/jsxgraphcore.js"></script>
</head>
<body>
<h1>Total Areas of Alternating Subtriangles in a n-gon with n=2m</h1>
Let P be a point connected to and inside the vertices of a 2m-gon.
Number the triangles counterclockwise from 1 to 2n. Then the sum of the areas of the even-numbered triangles is equal to the sum of the areas
of the odd-numbered triangles.<br>
Drag the point P to change the figure or change with the radio buttons the number of vertices.
<br><br>
<form>
<table><tr>
<td>n = </td>
<td><input type="radio" name="n" value="sechs" onClick="run(6);"> 6</td>
<td><input type="radio" name="n" value="acht" onClick="run(8);"> 8</td>
<td><input type="radio" name="n" value="zehn" checked="checked" onClick="run(10);"> 10</td>
</tr></table>
</form>
<table><tr><td>
<table border='1'>
<tr id="input1">
</tr>
<tr id="input2">
</tr>
</table>
</td>
<td width='100'></td>
<td>
<table border='1'>
<tr id="input3">
<td bgcolor='#3CB371' id="input3a"></td>
<td bgcolor='#DA70D6' id="input3b"></td>
</tr>
<tr>
<td align='center' id='sum_1'> 1 </td>
<td align='center' id='sum_2'> 2 </td>
</tr>
</table>
</td>
</table>
<br>
<!-- Drawing area -->
<div id="box" class="jxgbox" style="width:800px; height:400px; overflow:hidden; /*position:absolute; top:0px; left:0px;*/"></div>
<!-- Drawing area -->
<div id="debug" style="display:block;"></div>
<script type="text/javascript">
/* <![CDATA[ */
var board = JXG.JSXGraph.initBoard('box', {originX: 400, originY: 200, unitX: 150, unitY: 150});
function run(n) {
// Tabellen oberhalb der Zeichenflaeche abhaengig von n anlegen
var tr1=document.getElementById("input1");
while(tr1.firstChild) {
var node = tr1.firstChild;
var away = tr1.removeChild(node);
}
for(var i = 1; i <= n; i++) {
if(i % 2 != 0) {
var neu = document.createElement("td");
var neuText = document.createTextNode("A");
var neu2 = document.createElement("sub");
var neuText2 = document.createTextNode(i);
neu2.appendChild(neuText2);
neu.appendChild(neuText);
neu.appendChild(neu2);
tr1.appendChild(neu);
neu.style.backgroundColor = '#3CB371';
}
else {
var neu = document.createElement("td");
var neuText = document.createTextNode("A");
var neu2 = document.createElement("sub");
var neuText2 = document.createTextNode(i);
neu2.appendChild(neuText2);
neu.appendChild(neuText);
neu.appendChild(neu2);
tr1.appendChild(neu);
neu.style.backgroundColor = '#DA70D6';
}
}
var tr2=document.getElementById("input2");
while(tr2.firstChild) {
var node = tr2.firstChild;
var away = tr2.removeChild(node);
}
for(var i = 1; i <= n; i++) {
var neu = document.createElement("td");
neu.setAttribute("id","area_"+i);
var neuText = document.createTextNode("1");
neu.appendChild(neuText);
tr2.appendChild(neu);
}
var tr3a=document.getElementById("input3a");
var s = "";
for(var i = 1; i <= n; i=i+2) {
if(i != 1) {
s += " + A<sub>"+i+"</sub>";
}
else {
s += "A<sub>"+i+"</sub>";
}
}
tr3a.innerHTML = s;
var tr3b=document.getElementById("input3b");
s = "";
for(var i = 2; i <= n; i=i+2) {
if(i != 2) {
s += " + A<sub>"+i+"</sub>";
}
else {
s += "A<sub>"+i+"</sub>";
}
}
tr3b.innerHTML = s;
// Board erzeugen
JXG.JSXGraph.freeBoard(board);
board = JXG.JSXGraph.initBoard('box', {originX: 400, originY: 200, unitX: 150, unitY: 150});
//board.renderer.suspendRedraw();
var r = 1;
// n-Eck
var p = new Array(n);
for(var i = 0; i < n; i++) {
p[i] = board.create('point', [r*Math.sin(i*2*Math.PI/n), r*Math.cos(i*2*Math.PI/n)] , {fixed:true,fillColor:'blue'});
}
// Zieh-Punkt im Inneren
var q = board.create('point',[-0.3,-0.3],{fillColor:'red',strokeColor:'red',name:'P'});
// Polygone
var polygones = new Array(n);
var mp = new Array(n);
for(var i=0; i<n; i++) {
if(i < n-1) {
if(i % 2 == 0) {
polygones[i] = board.create('polygon',[q,p[i],p[i+1]], {fillColor:'#3CB371',name:''});
}
else {
polygones[i] = board.create('polygon',[q,p[i],p[i+1]], {fillColor:'#DA70D6',name:''});
}
// Labels fuer die Polygone....
mp[i] = board.create('circumcircleMidpoint',[q,p[i],p[i+1]], {name:'',visible:false});
board.create('text',[
(function(j){ return function(){ return mp[j].X();}; })(i),
(function(j){ return function(){ return mp[j].Y();}; })(i),
'A<sub>'+(i+1)+'</sub>'
],{});
}
else {
polygones[i] = board.create('polygon',[q,p[i],p[0]], {fillColor:'#DA70D6',name:''});
// Labels fuer die Polygone....
mp[i] = board.create('circumcircleMidpoint',[q,p[i],p[0]], {name:'',visible:false});
board.create('text',[
(function(j){ return function(){ return mp[j].X();}; })(i),
(function(j){ return function(){ return mp[j].Y();}; })(i),
'A<sub>'+(i+1)+'</sub>'
],{});
}
}
// Labels der Polygonseiten verstecken
/*
for(var j=0; j < n; j++) {
for(var i=0; i < polygones[j].borders.length; i++) {
polygones[j].borders[i].label.content.hideElement();
}
}
*/
// Hook-Funktion definieren, die die Tabelleninhalte updatet
function print_table(board) {
var distances = new Array(n);
for(var i=0; i<n; i++) {
distances[i] = q.coords.distance(JXG.COORDS_BY_USER, p[i].coords);
}
var border_poly = 2*r*Math.sin(Math.PI/n);
var areas = new Array(n);
for(var i=0; i<n; i++) {
if(i<n-1) {
var bordersum = (border_poly+distances[i]+distances[i+1])/2.0;
areas[i] = Math.sqrt(bordersum*(bordersum-border_poly)*(bordersum-distances[i])*(bordersum-distances[i+1]));
}
else {
var bordersum = (border_poly+distances[i]+distances[0])/2.0;
areas[i] = Math.sqrt(bordersum*(bordersum-border_poly)*(bordersum-distances[i])*(bordersum-distances[0]));
}
}
for(var i=0; i<n; i++) {
$('area_'+(i+1)).innerHTML = areas[i].toFixed(2);
}
var sum = 0;
for(var i=0; i<n; i=i+2) {
sum += areas[i];
}
$('sum_2').innerHTML = sum.toFixed(4);
sum = 0;
for(var i=1; i<n; i=i+2) {
sum += areas[i];
}
$('sum_1').innerHTML = sum.toFixed(4);
}
//board.renderer.unsuspendRedraw();
// und beim board registrieren.
print_table_id = board.addHook(print_table);
}
run(10);
/* ]]> */
</script>
</body>
</html>
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