Math E-320: Spring 2013
Teaching Math with a Historical Perspective
Mathematics E-320:
Instructor: Oliver Knill
Office: SciCtr 432

## Special Points for Circles

Centroid
```(* Proof that the centroid exists, Oliver Knill, January 13, 2010  *)
a={aa,a2,a2};b={b1,b2,b3};c={c1,c2,c3};u=(b+c)/2;v=(a+c)/2;w=(a+b)/2;
p=(a+b+c)/3;          Simplify[p == a/3+2u/3== b/3+ 2v/3 == c/3+2w/3]
```
Orthocenter
```(* Proof that the orthocenter exists, Oliver Knill, January 13, 2010  *)
a={a1,a2}; b={b1,b2}; c={c1,c2};
f[{a_,b_,c_}]:=a+ (b-a)*((c-a).(b-a))/((b-a).(b-a));
m=f[{b,c,a}]; n=f[{c,a,b}]; p=f[{a,b,c}];
L[x_,y_,z_,w_]:=x+(t/. Solve[x+t(y-x)==z+s(w-z),{t,s}][[1]])(y-x);
Simplify[ L[a,m,b,n]==L[a,m,c,p]==L[b,n,c,p] ]
```
Center of Circumscribed Circle
```(* Proof that center of the circumscribed circle  exists, Oliver Knill, January 13, 2010  *)
a={a1,a2}; b={b1,b2}; c={c1,c2}; u1=(b+c)/2; v1=(a+c)/2; w1=(a+b)/2;
p[x_]:={-x[[2]],x[[1]]}; u2=u1+p[c-b]; v2=v1+p[c-a]; w2=w1+p[a-b];
L[x_,y_,z_,w_]:=x+(t/. Solve[x+t(y-x)==z+s(w-z),{t,s}][[1]])(y-x);
Simplify[ L[u1,u2,v1,v2]==L[u1,u2,w1,w2]==L[v1,v2,w1,w2] ]
```
Center of Inscribed Circle
```(* Proof that center of the inscribed circle  exists, Oliver Knill, January 13, 2010  *)
a={a1,a2}; b={b1,b2}; c={c1,c2};
f[a_,b_,c_] := a+ ((b-a)/Sqrt[(b-a).(b-a)] +(c-a)/Sqrt[(c-a).(c-a)])/2
u=f[a,b,c]; v=f[b,c,a]; w=f[c,a,b];
L[x_,y_,z_,w_]:=x+(t/. Solve[x+t(y-x)==z+s(w-z),{t,s}][[1]])(y-x);
Simplify[ L[a,u,b,v]==L[a,u,c,w]==L[b,v,c,w] ]
```
Feuerbach proof
```(* Proof that the 9 point circle exists, Oliver Knill, January 13, 2010  *)

LineIntersect[A1_,A2_,B1_,B2_]:= A1 + (t /. Solve[A1+t (A2-A1) == B1+s (B2-B1),{t,s}][[1]]) (A2-A1);
BasePoint[a_,b_,c_]:=a+(b-a) ((c-a).(b-a)/((b-a).(b-a)));      MidPoint[a_,b_]:=(a+b)/2;
R = 2*(x3*(y1-y2)+x1*(y2-y3) + x2*(-y1 + y3));
n1=(x3^2*(y1-y2)+(x1^2+(y1-y2)*(y1-y3))*(y2-y3)+x2^2*(-y1+y3))/R;
n2=(-(x2^2*x3)+x1^2*(-x2+x3)+x3*(y1^2-y2^2)+x1*(x2^2-x3^2+y2^2-y3^2)+x2*(x3^2 - y1^2+y3^2))/R;

p1={px1,py1}; p2={px2,py2}; p3={px3,py3};
m3 = MidPoint[p1,p2]; m2 =MidPoint[p1,p3]; m1 = MidPoint[p2,p3];
h3 = BasePoint[p1,p2,p3]; h1=BasePoint[p2,p3,p1]; h2=BasePoint[p3,p1,p2];
m  = LineIntersect[p1,h1,p2,h2]; q1=MidPoint[m,p1]; q2=MidPoint[m,p2]; q3=MidPoint[m,p3];
Simplify[Umkreis[q1,q2,q3]==Umkreis[m1,m2,m3]==Umkreis[h1,h2,h3]]
```
Please send questions and comments to knill@math.harvard.edu
Math E320| Oliver Knill | Spring 2013 | Extension School | Harvard University