quadrilateral circumscribed around a circle

Question

Given: Quadrilateral WXYZ with lines XY, YZ, ZW, and XW tangent to circle K. WXYZ is a trapeziod.

 

To Prove: XY + WZ = XW + YZ

Answer 1

     This works for ANY quadrilateral and uses the fact that the two segments drawn from the same external point that are tangent to a circle must be equal. Let A be the point of tangency for WX, B be the point of tangency for XY, C for YZ, and D for WZ. Using the theorem above

 

XA=XB

YC=YB

ZC=ZD

WA=WD

 

So the sum of these four equations must also be equal. That is

XA+YC+ZC+WA = XB+YB+ZD+WD.       XA+WA=WX,            YC+ZC=YZ,        XB+YB=XY,     and ZD+WD=WZ   replacing we get

WX+YZ               = XY+WZ

 

Exactly what you wanted to prove.