Distance[{x1_,y1_,x2_,y2_}]:=Sqrt[(x1-x2)^2+(y1-y2)^2]
OnCircle[{x1_,y1_},{x2_,y2_},u_]:=Sqrt[(x1-x2)^2+(y1-y2)^2]==u
Perpendicular[{x1_,y1_},{x2_,y2_},{x3_,y3_},{x4_,y4_}]:=(x1-x2)*(x3-x4)+(y1-y2)*(y3-y4)==0
Collinear[{x1_,y1_},{x2_,y2_},{x3_,y3_}]:=(x1-x2)*(y2-y3)==(x2-x3)*(y1-y2)
Parallel[{x1_,y1_},{x2_,y2_},{x3_,y3_},{x4_,y4_}]:=(x1-x2)*(y3-y4)==(x3-x4)*(y1-y2)
Simetric[{x1_,y1_},{x2_,y2_},{x3_,y3_}]:=(x1+x3)/2==x2&&(y1+y3)/2==y2
A={0,0};M={1,0};P={c,d};B={a,b};Q={r,s};X={x,y};
OnCircle[Q,A,R]
Parallel[B,X,P,Q]
Perpendicular[A,M,X,Q]
ec1=Eliminate[Sqrt[ r^2+s^2]== R&&(d-s)(a-x)==(c-r)(b-y)&&r-x==0,{r,s}]
Solve[ec1,y]

Factor[(a^2) (c^2) (R^2)-2 (a^2) c (R^2) x-2 a (c^2) (R^2) x-(a^2) (c^2) (x^2)+(a^2) (R^2) (x^2)+4 a c (R^2) (x^2)+(c^2) (R^2) (x^2)+2 (a^2) c (x^3)+2 a (c^2) (x^3)-2 a (R^2) (x^3)-2 c (R^2) (x^3)-(a^2) (x^4)-4 a c (x^4)-(c^2) (x^4)+(R^2) (x^4)+2 a (x^5)+2 c (x^5)-(x^6)]