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Tangents on circle
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<iframe 
    src="http://jsxgraph.uni-bayreuth.de/share/iframe/tangents-on-circle" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Tangents on circle" 
    allowfullscreen
></iframe>
This code has to 
<div id="board-0-wrapper" class="jxgbox-wrapper " style="width: 100%; ">
   <div id="board-0" class="jxgbox" style="aspect-ratio: 1 / 1; width: 100%;" data-ar="1 / 1"></div>
</div>

<script type = "text/javascript"> 
    /*
    This example is licensed under a 
    Creative Commons Attribution ShareAlike 4.0 International License.
    https://creativecommons.org/licenses/by-sa/4.0/
    
    Please note you have to mention 
    The Center of Mobile Learning with Digital Technology
    in the credits.
    */
    
    const BOARDID = 'board-0';

    var board = JXG.JSXGraph.initBoard(BOARDID, { boundingbox: [-5, 5, 5, -5], keepaspectratio: true });
    
    // Set some point attributes.
    board.options.point.strokeColor = 'red';
    board.options.point.fillColor = 'red';
    board.options.point.strokeOpacity = 0.8;
    board.options.point.fillOpacity = 0.8;
    
    var m = board.create('point', [0.5, 0.3]);
    var b = board.create('point', [3, 0]);
    var c = board.create('circle', [m, b]);
    
    var p = board.create('point', [-4, 2], {name: 'p', label: {autoPosition: true}});
    
    // If p is not on c, the tangent is the polar.
    var t = board.create('tangent', [c, p], { name: 'polar', withLabel: true, label: {position: '0.7 left', distance: 0.5} });
    
    var i1 = board.create('intersection', [c, t, 0], { visible: false });
    var i2 = board.create('intersection', [c, t, 1], { visible: false });
    var t1 = board.create('tangent', [c, i1]);
    var t2 = board.create('tangent', [c, i2]);
 </script> 
/*
This example is licensed under a 
Creative Commons Attribution ShareAlike 4.0 International License.
https://creativecommons.org/licenses/by-sa/4.0/

Please note you have to mention 
The Center of Mobile Learning with Digital Technology
in the credits.
*/

const BOARDID = 'your_div_id'; // Insert your id here!

var board = JXG.JSXGraph.initBoard(BOARDID, { boundingbox: [-5, 5, 5, -5], keepaspectratio: true });

// Set some point attributes.
board.options.point.strokeColor = 'red';
board.options.point.fillColor = 'red';
board.options.point.strokeOpacity = 0.8;
board.options.point.fillOpacity = 0.8;

var m = board.create('point', [0.5, 0.3]);
var b = board.create('point', [3, 0]);
var c = board.create('circle', [m, b]);

var p = board.create('point', [-4, 2], {name: 'p', label: {autoPosition: true}});

// If p is not on c, the tangent is the polar.
var t = board.create('tangent', [c, p], { name: 'polar', withLabel: true, label: {position: '0.7 left', distance: 0.5} });

var i1 = board.create('intersection', [c, t, 0], { visible: false });
var i2 = board.create('intersection', [c, t, 1], { visible: false });
var t1 = board.create('tangent', [c, i1]);
var t2 = board.create('tangent', [c, i2]);
<jsxgraph width="100%" aspect-ratio="1 / 1" title="Tangents on circle" description="This construction was copied from JSXGraph examples database: BTW HERE SHOULD BE A GENERATED LINKuseGlobalJS="false">
   /*
   This example is licensed under a 
   Creative Commons Attribution ShareAlike 4.0 International License.
   https://creativecommons.org/licenses/by-sa/4.0/
   
   Please note you have to mention 
   The Center of Mobile Learning with Digital Technology
   in the credits.
   */
   
   var board = JXG.JSXGraph.initBoard(BOARDID, { boundingbox: [-5, 5, 5, -5], keepaspectratio: true });
   
   // Set some point attributes.
   board.options.point.strokeColor = 'red';
   board.options.point.fillColor = 'red';
   board.options.point.strokeOpacity = 0.8;
   board.options.point.fillOpacity = 0.8;
   
   var m = board.create('point', [0.5, 0.3]);
   var b = board.create('point', [3, 0]);
   var c = board.create('circle', [m, b]);
   
   var p = board.create('point', [-4, 2], {name: 'p', label: {autoPosition: true}});
   
   // If p is not on c, the tangent is the polar.
   var t = board.create('tangent', [c, p], { name: 'polar', withLabel: true, label: {position: '0.7 left', distance: 0.5} });
   
   var i1 = board.create('intersection', [c, t, 0], { visible: false });
   var i2 = board.create('intersection', [c, t, 1], { visible: false });
   var t1 = board.create('tangent', [c, i1]);
   var t2 = board.create('tangent', [c, i2]);
</jsxgraph>

Tangents on circle

Geometry
This JSXGraph demo shows the __polar of a point__ with respect to a circle. If point *p* lies outside circle *c*, the polar line is constructed with the two tangents from $p$ to the circle. The intersections of the tangents with the circle define two points which define the *polar line*. If $p$ is on the circle, the polar line is tangent to the circle through $p$. If you look at the code, the construction is the other way round: ``` t = board.create('tangent', [c, p]); ``` constructs the polar line of $p$. The intersection points of the polar line $t$ with the circle are those points which define the tangent line through the point $p$, i.e. where the circle is touched by the tangents to the circle through $p$.
// Define the id of your board in BOARDID

var board = JXG.JSXGraph.initBoard(BOARDID, { boundingbox: [-5, 5, 5, -5], keepaspectratio: true });

// Set some point attributes.
board.options.point.strokeColor = 'red';
board.options.point.fillColor = 'red';
board.options.point.strokeOpacity = 0.8;
board.options.point.fillOpacity = 0.8;

var m = board.create('point', [0.5, 0.3]);
var b = board.create('point', [3, 0]);
var c = board.create('circle', [m, b]);

var p = board.create('point', [-4, 2], {name: 'p', label: {autoPosition: true}});

// If p is not on c, the tangent is the polar.
var t = board.create('tangent', [c, p], { name: 'polar', withLabel: true, label: {position: '0.7 left', distance: 0.5} });

var i1 = board.create('intersection', [c, t, 0], { visible: false });
var i2 = board.create('intersection', [c, t, 1], { visible: false });
var t1 = board.create('tangent', [c, i1]);
var t2 = board.create('tangent', [c, i2]);

license

This example is licensed under a Creative Commons Attribution ShareAlike 4.0 International License.
Please note you have to mention The Center of Mobile Learning with Digital Technology in the credits.