Sadovs Theorem


Exhibit: table of content

Mathematics Maths21a, Summer 2005
Multivariable Calculus
Oliver Knill, SciCtr 434, knill.harvard.edu
Sergey Sadov proved in 2004 the following theorem in elementary geometry.

THEOREM: If ABCD is a quadrilateral on a circle, then
           |AB| |BC| |CA| + |AC| |CD| |DA|  = |BC| |CD| |DB| + |AB| |BD| |DA|
           
Proof. (This is a Mathematica version of a Maple proof found by Shalosh B. Ekhad, the computer collaborator of Doron Zeilberg at Rutgers university).
        
           H[A_,B_]:=(A[[1]]-B[[1]])^2+(A[[2]]-B[[2]])^2;
           P[t_]:={t+1/t,(t-1/t)/I}/2;
           T[A_,B_,C_,D_]:=((A+B-C-D)^2-4*(A*B+C*D))^2-64*A*B*C*D;
           U[A_,B_,C_]:=H[A,B]*H[B,C]*H[C,A];
           K={P[t[1]],P[t[2]],P[t[3]],P[t[4]]};
           S=T[U[P[t[2]],P[t[3]],P[t[4]]],U[P[t[1]],P[t[3]],P[t[4]]],
               U[P[t[1]],P[t[2]],P[t[4]]],U[P[t[1]],P[t[2]],P[t[3]]]]==0; 
           Simplify[S]
           
Side remark: Ekhads Maple Proof uses a 360 Character Maple code, which proves it in 0.05 seconds. The above Mathematica code has 336 characters and needed 0.01 seconds to complete on a Pentium D 840.
Please send comments to maths21a.harvard.edu
Oliver Knill, Maths21a, Multivariable Calculus, Summer 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University