Let http://data.artofproblemsolving.com/...ccba2ee9f0.gif be a line segment in the plane and http://data.artofproblemsolving.com/...29b6d953ec.gif a points between http://data.artofproblemsolving.com/...63d9131c1b.gif and http://data.artofproblemsolving.com/...e18289357d.gif. Construct isosceles triangles http://data.artofproblemsolving.com/...7f2836d548.gif and http://data.artofproblemsolving.com/...3d43065669.gif on one side of the segment http://data.artofproblemsolving.com/...ccba2ee9f0.gif such that http://data.artofproblemsolving.com/...43648fb3cc.gif and an isosceles triangle http://data.artofproblemsolving.com/...0c4c6b2905.gif on the other side of http://data.artofproblemsolving.com/...ccba2ee9f0.gif such that http://data.artofproblemsolving.com/...2ab5396490.gif

show that $\displaystyle AP+CQ-AR = 0 $