Proof of the Inscribed Angle Theorem - the vertex of central angle is outside a inscribed angle

						var board = JXG.JSXGraph.initBoard('box1', {
boundingbox: [-10, 10, 10, -10],showNavigation:false, showcopyright: false, axis: false,
});

var p1 = board.create('point', [0, 2.0],{name:'O',face:'',label:{offset:[-15,0]}});
var c1 = board.create('circle', [p1, 6],);
var g1 = board.create('glider', [1.5, 4.5, c1],{face:'',label:{offset:[5,5]}});

var g2 = board.create('glider', [4.0, 3, c1],{face:'',label:{offset:[5,5]}});

var g3 = board.create('glider', [1, -2, c1],{face:'',label:{offset:[0,-10]}});

var l1 = board.create('segment', [g1,g3],);


var l2 = board.create('segment', [g3,g2]);

var l3 = board.create('segment', [p1,g1]);
var l4 = board.create('segment', [p1,g2]);
var l5 = board.create('line', [p1,g3],{visible:false});
var i1 = board.create('intersection', [c1, l5, 1],{face:'',label:{offset:[-5,10]}});
var l6 = board.create('segment', [i1,g3]);

var a1 = board.create('angle', [g2, g3, g1],{name:'α',radius:3,label:{offset:[-4,4]}});

var a2 = board.create('angle', [g1, g3, i1],{name:'α2', radius:3, label:{offset:[-7,2]}});

var b1 = board.create('angle', [g2, p1, g1],{name:'β',radius:2,label:{offset:[-14,-4]}});

var b2 = board.create('angle', [g1, p1, i1],{name:'β2',radius:2,label:{offset:[-5,2]}});

The image appears in the following posts

  1. In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. A central angle is an angle whose vertex is the center of the circle and whose arms intersect the circumference of the circle.

    The inscribed angle theorem tells us if an inscribed angle and a central angle of a circle intercept the same arc, the inscribed angle is the half of the central angle.

    To prove the theorem, consider 3 cases when the vertex is on one arm of the inscribed angle, the vertex is between the arms of the inscribed angle and the vertex is outside the inscribed angle.

Triangle posted 1 year ago

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