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 Th61:
  A,B,C is_a_triangle implies median(A,B,C) is being_line
  proof
    assume
A1: A,B,C is_a_triangle;
A2: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
    assume not median(A,B,C) is being_line;
    then consider x such that
A3: median(A,B,C)={x} by Th6;
    set D = the_midpoint_of_the_segment(B,C);
    A in median(A,B,C) & D in median(A,B,C) by EUCLID_4:41;
    then A = x & D = x by A3,TARSKI:def 1;
    then A in LSeg(B,C) & LSeg(B,C) c= Line(B,C) by Th21,RLTOPSP1:73;
    hence contradiction by A1,A2,MENELAUS:13;
  end;
