reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

theorem Th59:
  A,B,C is_a_triangle &
  D = (1-1/2) * B + 1/2 * C &
  E = (1-1/2) * C + 1/2 * A &
  F = (1-1/2) * A + 1/2 * B
  implies Line(A,D),Line(B,E),Line(C,F) are_concurrent
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: D = (1-1/2) * B + 1/2 * C and
A3: E = (1-1/2) * C + 1/2 * A and
A4: F = (1-1/2) * A + 1/2 * B;
    set lambda = 1/2;
    set mu = 1/2;
    set nu = 1/2;
    (lambda/(1-lambda)) * (mu/(1-mu)) * (nu/(1-nu)) =1;
    hence thesis by A1,A2,A3,A4,MENELAUS:22;
  end;
