SimsonLineEulerCircumference::usage="The Simson line is the one that contains the projections of the 3 feet of the heights of a triangle to the point P over the circumference. Euler's circumference was added to observe that the midpoint of the line that goes from the orthocenter to the point P on the circumscribed circumference is always a point above the Euler circle. Reference: Wolfram Math World, Simson line; Personal. ";Animatex1=-2;y1=5;x2=7;y2=12;x3=9;y3=-4;h=;k=;r=;s1=Solve[{(y1-y2)x+(x2-x1)y-x2y1+x1y2==0,(y2-y1)y+(x2-x1)x-(y2-y1)rSin[θ]-(x2-x1)rCos[θ]-h(x2-x1)-k(y2-y1)==0},{x,y}];xp11=x/.s1;xp1=xp11.{1};yp11=y/.s1;yp1=yp11.{1};s2=Solve[{(y2-y3)x+(x3-x2)y-x3y2+x2y3==0,(y3-y2)y+(x3-x2)x-(y3-y2)rSin[θ]-(x3-x2)rCos[θ]-h(x3-x2)-k(y3-y2)==0},{x,y}];xp22=x/.s2;xp2=xp22.{1};yp22=y/.s2;yp2=yp22.{1};s3=Solve[{(y1-y3)x+(x3-x1)y-x3y1+x1y3==0,(y3-y1)y+(x3-x1)x-(y3-y1)rSin[θ]-(x3-x1)rCos[θ]-h(x3-x1)-k(y3-y1)==0},{x,y}];xp33=x/.s3;xp3=xp33.{1};yp33=y/.s3;yp3=yp33.{1};g1=Line[{{x1,y1},{x2,y2},{x3,y3},{x1,y1}}];g2=InfiniteLine[{{xp1,yp1},{xp2,yp2}}];g3=Circle[{h,k},r];g4=Line[{{xp1,yp1},{rCos[θ]+h,rSin[θ]+k}}];g5=Line[{{xp2,yp2},{rCos[θ]+h,rSin[θ]+k}}];g6=Line[{{xp3,yp3},{rCos[θ]+h,rSin[θ]+k}}];g7=Point[{rCos[θ]+h,rSin[θ]+k}];g8=Point[{xp1,yp1}];g9=Point[{xp2,yp2}];g10=Point[{xp3,yp3}];g11=Circle,,;g12=Point,;g13=Point,;g14=Line,,{rCos[θ]+h,rSin[θ]+k};Show[{Graphics[{Blue,g1}],Graphics[{Red,g2}],Graphics[{LightBrown,g3}],Graphics[{LightRed,g4}],Graphics[{LightRed,g5}],Graphics[{LightRed,g6}],Graphics[{Black,g7}],Graphics[{Black,g8}],Graphics[{Black,g9}],Graphics[{Black,g10}],Graphics[{LightBlue,g11}],Graphics[{Black,g12}],Graphics[{Black,g13}],Graphics[{Green,g14}]},AxesTrue,AxesOrigin{0,0}],{θ,0,2π,0.1},AnimationRunningFalse
488
79
298
79
65
101
79
309
79
729
158
426725
24964
130
79
431
79
rCos[θ]+h+
130
79
2
rSin[θ]+k+
431
79
2
130
79
431
79
Out[]=
