Circum circles of subtriangles: Difference between revisions

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Let ABC be a triangle and let the incircle intersect <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively.
Let the circumcircles of <math>AB'C'</math>, <math>A'BC'</math>, and <math>A'B'C</math> intersect the circumcircle of <math>ABC</math> (apart from <math>A</math>, <math>B</math>, and <math>C</math>) at <math>A''</math>, <math>B''</math>, and <math>C''</math>, respectively.
Then <math>A'A''</math>, <math>B'B''</math>, and <math>C'C''</math> meet in one point, <math>P</math>.
<jsxgraph width="800" height="600">
<jsxgraph width="800" height="600">
brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-1,4,12,-4], keepaspectratio:true});
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'});
p1 = brd.create('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'});
p2 = brd.create('point', [7.5,0.5] , {name:'B',fillColor:'red',strokeColor:'red'});
p3 = brd.createElement('point', [2,3] , {name:'C',fillColor:'red',strokeColor:'red'});
p3 = brd.create('point', [2,3] , {name:'C',fillColor:'red',strokeColor:'red'});
      
      
b1 = brd.createElement('line',['A','B'],{name:'',straightFirst:false,straightLast:false});
b1 = brd.create('segment',['A','B'],{name:''});
b2 = brd.createElement('line',['A','C'],{name:'',straightFirst:false,straightLast:false});
b2 = brd.create('segment',['A','C'],{name:''});
b3 = brd.createElement('line',['C','B'],{name:'',straightFirst:false,straightLast:false});
b3 = brd.create('segment',['C','B'],{name:''});
      
      
c1 = brd.createElement('circumcircle',['A','B','C'],{name:''});
c1 = brd.create('circumcircle',['A','B','C'],{name:'', point: {visible: false}});
c1[1].setProperty('strokeColor:#AAAAAA');  
c1.setProperty('strokeColor:#AAAAAA');  
c1[0].hideElement(); // hide center of circle
      
      
l1 = brd.createElement('bisector',['B','A','C'],{name:'',visible:false}); // alpha
l1 = brd.create('bisector',['B','A','C'],{name:'',visible:false}); // alpha
l2 = brd.createElement('bisector',['C','B','A'],{name:'',visible:false}); // beta
l2 = brd.create('bisector',['C','B','A'],{name:'',visible:false}); // beta
      
      
i1 = brd.createElement('intersection',[l1,l2,0],{name:'',visible:false});
i1 = brd.create('intersection',[l1,l2,0],{name:'',visible:false});
pp1 = brd.createElement('perpendicularpoint',[i1,b1],{name:"C'",fillColor:'blue'});
pp1 = brd.create('perpendicularpoint',[i1,b1],{name:"C'",fillColor:'blue'});
pp2 = brd.createElement('perpendicularpoint',[i1,b2],{name:"B'",fillColor:'blue'});
pp2 = brd.create('perpendicularpoint',[i1,b2],{name:"B'",fillColor:'blue'});
pp3 = brd.createElement('perpendicularpoint',[i1,b3],{name:"A'",fillColor:'blue'});
pp3 = brd.create('perpendicularpoint',[i1,b3],{name:"A'",fillColor:'blue'});
      
      
c2 = brd.createElement('circumcircle',[pp1,pp2,pp3],{name:''});
c2 = brd.create('circumcircle',[pp1,pp2,pp3],{name:'', strokeColor: '#3CB371', point: {visible: false}});
c2[1].setProperty('strokeColor:#3CB371');
c3 = brd.create('circumcircle',[p3,pp2,pp3],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
c2[0].hideElement();
c4 = brd.create('circumcircle',[p2,pp1,pp3],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
c5 = brd.create('circumcircle',[p1,pp2,pp1],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
      
      
c3 = brd.createElement('circumcircle',[p3,pp2,pp3],{name:''});
i2 = brd.create('otherintersection',[c3,c1,p3],{name:"C''",fillColor:'blue'});
c3[1].setProperty('strokeColor:#FF8C00');
i3 = brd.create('otherintersection',[c4,c1,p2],{name:"B''",fillColor:'blue'});
c3[0].hideElement();
i4 = brd.create('otherintersection',[c5,c1,p1],{name:"A''",fillColor:'blue'});  


c4 = brd.createElement('circumcircle',[p2,pp1,pp3],{name:''});
ll1 = brd.create('segment',[i2,pp1],{name:'',strokeColor:'#FF6347'});
c4[1].setProperty('strokeColor:#FF8C00');
ll2 = brd.create('segment',[i3,pp2],{name:'',strokeColor:'#FF6347'});
c4[0].hideElement();
ll3 = brd.create('segment',[i4,pp3],{name:'',strokeColor:'#FF6347'});
   
i5 = brd.create('intersection',[ll1,ll2,0],{name:"P",fillColor:'#9932CC',strokeColor:'#9932CC'});
brd.update();
</jsxgraph>


c5 = brd.createElement('circumcircle',[p1,pp2,pp1],{name:''});
===The underlying JavaScript code===
c5[1].setProperty('strokeColor:#FF8C00');
<source lang="javascript">
c5[0].hideElement();
brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-1,4,12,-4], keepaspectratio:true});
p1 = brd.create('point', [0.5,-1.5] , {name:'A',fillColor:'red',strokeColor:'red'});
p2 = brd.create('point', [7.5,0.5] , {name:'B',fillColor:'red',strokeColor:'red'});
p3 = brd.create('point', [2,3] , {name:'C',fillColor:'red',strokeColor:'red'});
   
b1 = brd.create('segment',['A','B'],{name:''});
b2 = brd.create('segment',['A','C'],{name:''});
b3 = brd.create('segment',['C','B'],{name:''});
   
c1 = brd.create('circumcircle',['A','B','C'],{name:'', point: {visible: false}});
c1.setProperty('strokeColor:#AAAAAA');
   
l1 = brd.create('bisector',['B','A','C'],{name:'',visible:false}); // alpha
l2 = brd.create('bisector',['C','B','A'],{name:'',visible:false}); // beta
   
i1 = brd.create('intersection',[l1,l2,0],{name:'',visible:false});
pp1 = brd.create('perpendicularpoint',[i1,b1],{name:"C'",fillColor:'blue'});
pp2 = brd.create('perpendicularpoint',[i1,b2],{name:"B'",fillColor:'blue'});
pp3 = brd.create('perpendicularpoint',[i1,b3],{name:"A'",fillColor:'blue'});
      
      
i2 = brd.createElement('otherintersection',[c3[1],c1[1],p3],{name:"C''",fillColor:'blue'});
c2 = brd.create('circumcircle',[pp1,pp2,pp3],{name:'', strokeColor: '#3CB371', point: {visible: false}});
i3 = brd.createElement('otherintersection',[c4[1],c1[1],p2],{name:"B''",fillColor:'blue'});
c3 = brd.create('circumcircle',[p3,pp2,pp3],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
i4 = brd.createElement('otherintersection',[c5[1],c1[1],p1],{name:"A''",fillColor:'blue'});  
c4 = brd.create('circumcircle',[p2,pp1,pp3],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
c5 = brd.create('circumcircle',[p1,pp2,pp1],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
   
i2 = brd.create('otherintersection',[c3,c1,p3],{name:"C''",fillColor:'blue'});
i3 = brd.create('otherintersection',[c4,c1,p2],{name:"B''",fillColor:'blue'});
i4 = brd.create('otherintersection',[c5,c1,p1],{name:"A''",fillColor:'blue'});  


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


[[Category:Examples]]
[[Category:Examples]]
[[Category:Geometry]]
[[Category:Geometry]]

Latest revision as of 09:56, 18 January 2021

Let ABC be a triangle and let the incircle intersect [math]\displaystyle{ BC }[/math], [math]\displaystyle{ CA }[/math], and [math]\displaystyle{ AB }[/math] at [math]\displaystyle{ A' }[/math], [math]\displaystyle{ B' }[/math], and [math]\displaystyle{ C' }[/math], respectively.

Let the circumcircles of [math]\displaystyle{ AB'C' }[/math], [math]\displaystyle{ A'BC' }[/math], and [math]\displaystyle{ A'B'C }[/math] intersect the circumcircle of [math]\displaystyle{ ABC }[/math] (apart from [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], and [math]\displaystyle{ C }[/math]) at [math]\displaystyle{ A'' }[/math], [math]\displaystyle{ B'' }[/math], and [math]\displaystyle{ C'' }[/math], respectively.

Then [math]\displaystyle{ A'A'' }[/math], [math]\displaystyle{ B'B'' }[/math], and [math]\displaystyle{ C'C'' }[/math] meet in one point, [math]\displaystyle{ P }[/math].

The underlying JavaScript code

brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-1,4,12,-4], keepaspectratio:true});
p1 = brd.create('point', [0.5,-1.5] , {name:'A',fillColor:'red',strokeColor:'red'});
p2 = brd.create('point', [7.5,0.5] , {name:'B',fillColor:'red',strokeColor:'red'});
p3 = brd.create('point', [2,3] , {name:'C',fillColor:'red',strokeColor:'red'});
    
b1 = brd.create('segment',['A','B'],{name:''});
b2 = brd.create('segment',['A','C'],{name:''});
b3 = brd.create('segment',['C','B'],{name:''});
    
c1 = brd.create('circumcircle',['A','B','C'],{name:'', point: {visible: false}});
c1.setProperty('strokeColor:#AAAAAA'); 
    
l1 = brd.create('bisector',['B','A','C'],{name:'',visible:false}); // alpha
l2 = brd.create('bisector',['C','B','A'],{name:'',visible:false}); // beta
    
i1 = brd.create('intersection',[l1,l2,0],{name:'',visible:false});
pp1 = brd.create('perpendicularpoint',[i1,b1],{name:"C'",fillColor:'blue'});
pp2 = brd.create('perpendicularpoint',[i1,b2],{name:"B'",fillColor:'blue'});
pp3 = brd.create('perpendicularpoint',[i1,b3],{name:"A'",fillColor:'blue'});
    
c2 = brd.create('circumcircle',[pp1,pp2,pp3],{name:'', strokeColor: '#3CB371', point: {visible: false}});
c3 = brd.create('circumcircle',[p3,pp2,pp3],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
c4 = brd.create('circumcircle',[p2,pp1,pp3],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
c5 = brd.create('circumcircle',[p1,pp2,pp1],{name:'', strokeColor: '#FF8C00', point: {visible: false}});
    
i2 = brd.create('otherintersection',[c3,c1,p3],{name:"C''",fillColor:'blue'});
i3 = brd.create('otherintersection',[c4,c1,p2],{name:"B''",fillColor:'blue'});
i4 = brd.create('otherintersection',[c5,c1,p1],{name:"A''",fillColor:'blue'}); 

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