# Difference between revisions of "Circum circles of subtriangles"

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'});