48-gon polygon cutting side

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

var p1 = board.create('point', [7, -1.0],{name:'O',face:''});
var p2 = board.create('point', [-7, -1.0],{face:'',label:{offset:[-15,-5]}});
var l1 = board.create('segment', [p1, p2]);
var t1 = board.create('transform', [-Math.PI/6, p1], {type: 'rotate'});

var p3 = board.create('point', [p2, t1], {color: 'blue',name:'',face:''},);

var l2 = board.create('line', [p1,p3],{visible:false});
var perp1 = board.create('perpendicular', [l1, p2],{visible:false});
var i1 = board.create('intersection', [perp1, l2],{face:''});
var l22 = board.create('segment', [p1, i1]);
var l3 = board.create('segment', [p2, i1]);
var bi1 = board.create('bisector', [i1, p1, p2],{visible:false});
var i2 = board.create('intersection', [perp1, bi1],{face:'',label:{offset:[-15,0]}});
var l4 = board.create('segment', [p1, i2]);
var bi2 = board.create('bisector', [i2, p1, p2],{visible:false});
var i3 = board.create('intersection', [perp1, bi2],{face:'',label:{offset:[-15,0]}});
var l4 = board.create('segment', [p1, i3]);
var bi3 = board.create('bisector', [i3, p1, p2],{visible:false});
var i4 = board.create('intersection', [perp1, bi3],{face:'',label:{offset:[-15,0]}});
var l5 = board.create('segment', [p1, i4]);

The image appears in the following posts

  1. Archimedes' approximation of Pi is by calculating the perimeter of regular polygons to determine the range of Pi. The more the number of sides, the more the perimeter of the polygon close to the circumference of its inscribed or circumscribed circle.

    Through doubling the sides of a hexagon to a 96-gon regular polygon, Archimedes found the ratio of the circumference of any circle to its diameter is less than 22/7 but greater than 223/71, i,.e.,

    \dfrac{223}{71} < \pi < \dfrac{22}{7}

    The lower limit of the approximation \dfrac{223}{71} is determined by the ratio of the perimeter of an inscribed 96-gon regular polygon to the diameter of its circumscribed circle. The upper limit \dfrac{22}{7} is determined by the ratio of a circumscribed 96-gon regular polygon to its inscribed circle. The method lies in 3 theorems. First, the ratio of the circumference of any circle to its diameter is constant. Second, the perimeter of an inscribed polygon is less than the circumference of its circumscribed circle. Third, the perimeter of a circumscribed polygon is greater than the circumference of its inscribed circle. So to prove the above approximation, implies proof of the following inequality,

    \dfrac{223}{71} < \dfrac{P_{96}}{R} <\pi < \dfrac{p_{96}}{r} < \dfrac{22}{7}

    Here we are going to skip the three preconditions, prove the above approximation directly.

Triangle posted 1 year ago

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