Circum circles of subtriangles

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Revision as of 12:18, 22 September 2009 by A WASSERMANN (talk | contribs)

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Let ABC be a triangle and let the incircle intersect BC, CA, and AB at A', B', and C', respectively.

Let the circumcircles of AB'C', A'BC', and A'B'C intersect the circumcircle of ABC (apart from A, B, and C) at A, B, and C, respectively.

Then A'A, B'B, and C'C meet in one point, P.

The underlying JavaScript code

brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-1,4,12,-4], keepaspectratio:true});
p1 = brd.createElement('point', [0.5,-1.5] , {name:'A',fillColor:'red',strokeColor:'red'});
p2 = brd.createElement('point', [7.5,0.5] , {name:'B',fillColor:'red',strokeColor:'red'});
p3 = brd.createElement('point', [2,3] , {name:'C',fillColor:'red',strokeColor:'red'});
    
b1 = brd.createElement('line',['A','B'],{name:'',straightFirst:false,straightLast:false});
b2 = brd.createElement('line',['A','C'],{name:'',straightFirst:false,straightLast:false});
b3 = brd.createElement('line',['C','B'],{name:'',straightFirst:false,straightLast:false});
    
c1 = brd.createElement('circumcircle',['A','B','C'],{name:''});
c1[1].setProperty('strokeColor:#AAAAAA'); 
c1[0].hideElement(); // hide center of circle
    
l1 = brd.createElement('bisector',['B','A','C'],{name:'',visible:false}); // alpha
l2 = brd.createElement('bisector',['C','B','A'],{name:'',visible:false}); // beta
    
i1 = brd.createElement('intersection',[l1,l2,0],{name:'',visible:false});
pp1 = brd.createElement('perpendicularpoint',[i1,b1],{name:"C'",fillColor:'blue'});
pp2 = brd.createElement('perpendicularpoint',[i1,b2],{name:"B'",fillColor:'blue'});
pp3 = brd.createElement('perpendicularpoint',[i1,b3],{name:"A'",fillColor:'blue'});
    
c2 = brd.createElement('circumcircle',[pp1,pp2,pp3],{name:''});
c2[1].setProperty('strokeColor:#3CB371');
c2[0].hideElement();
    
c3 = brd.createElement('circumcircle',[p3,pp2,pp3],{name:''});
c3[1].setProperty('strokeColor:#FF8C00');
c3[0].hideElement();

c4 = brd.createElement('circumcircle',[p2,pp1,pp3],{name:''});
c4[1].setProperty('strokeColor:#FF8C00');
c4[0].hideElement();

c5 = brd.createElement('circumcircle',[p1,pp2,pp1],{name:''});
c5[1].setProperty('strokeColor:#FF8C00');
c5[0].hideElement();
    
i2 = brd.createElement('otherintersection',[c3[1],c1[1],p3],{name:"C''",fillColor:'blue'});
i3 = brd.createElement('otherintersection',[c4[1],c1[1],p2],{name:"B''",fillColor:'blue'});
i4 = brd.createElement('otherintersection',[c5[1],c1[1],p1],{name:"A''",fillColor:'blue'}); 

ll1 = brd.createElement('line',[i2,pp1],{name:'',straightFirst:false,straightLast:false,strokeColor:'#FF6347'});
ll2 = brd.createElement('line',[i3,pp2],{name:'',straightFirst:false,straightLast:false,strokeColor:'#FF6347'});
ll3 = brd.createElement('line',[i4,pp3],{name:'',straightFirst:false,straightLast:false,strokeColor:'#FF6347'});
    
i5 = brd.createElement('intersection',[ll1,ll2,0],{name:"P",fillColor:'#9932CC',strokeColor:'#9932CC'});