# your code goes hereimport math;a,b,c=13.0,14.0,15.0;s=(a+b+c)/2;S=math.sqrt(s*(s-a)*(s-b)*(s-c));R_ABC=S/s;cosC=(a*a+b*b-c*c)/(2*a*b);sinC=math.sqrt(1-cosC**2);Ax,Ay=b*cosC,b*sinC;Bx,By=a,0.0;Cx,Cy=0.0,0.0;Dx,Dy=s-c,0.0;Ex,Ey=(s-c)/b*Ax,(s-c)/b*Ay;Fx,Fy=Ax+(s-a)/c*(Bx-Ax),Ay+(s-a)/c*(By-Ay);S_DEF=0.5*abs(Dx*(Ey-Fy)+Ex*(Fy-Dy)+Fx*(Dy-Ey));d=lambda x1,y1,x2,y2:math.sqrt((x1-x2)**2+(y1-y2)**2);DE=d(Dx,Dy,Ex,Ey);EF=d(Ex,Ey,Fx,Fy);FD=d(Fx,Fy,Dx,Dy);r_DEF=S_DEF/((DE+EF+FD)/2);proj=lambda Px,Py,Ux,Uy,Vx,Vy: (Ux+((Px-Ux)*(Vx-Ux)+(Py-Uy)*(Vy-Uy))/((Vx-Ux)**2+(Vy-Uy)**2)*(Vx-Ux), Uy+((Px-Ux)*(Vx-Ux)+(Py-Uy)*(Vy-Uy))/((Vx-Ux)**2+(Vy-Uy)**2)*(Vy-Uy));Kx,Ky=proj(Dx,Dy,Ex,Ey,Fx,Fy);Mx,My=proj(Ex,Ey,Fx,Fy,Dx,Dy);Nx,Ny=proj(Fx,Fy,Dx,Dy,Ex,Ey);S_KMN=0.5*abs(Kx*(My-Ny)+Mx*(Ny-Ky)+Nx*(Ky-My));print(f"Ratio={S_KMN/S:.6f} Hyp={(r_DEF/R_ABC)**4:.6f}")"

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