On the sides of triangle ABC, isosceles right-angled triangles AUB,CVB, and AWC are placed. These three triangles have their right angles at vertices U,V , and W, respectively. Triangle AUB lies completely inside triangle ABC and triangles CVB and AWC lie completely outside ABC. See the figure. Prove that quadrilateral UVCW is a parallelogram.
[asy]
import markers;unitsize(1.5 cm);pair A, B, C, U, V, W;A = (0,0);
B = (2,0);
C = (1.7,2.5);
U = (B + rotate(90,A)*(B))/2;
V = (B + rotate(90,C)*(B))/2;
W = (C + rotate(90,A)*(C))/2;draw(A--B--C--cycle);
draw(A--W, StickIntervalMarker(1,1,size=2mm));
draw(C--W, StickIntervalMarker(1,1,size=2mm));
draw(B--V, StickIntervalMarker(1,2,size=2mm));
draw(C--V, StickIntervalMarker(1,2,size=2mm));
draw(A--U, StickIntervalMarker(1,3,size=2mm));
draw(B--U, StickIntervalMarker(1,3,size=2mm));
draw(rightanglemark(A,U,B,5));
draw(rightanglemark(B,V,C,5));
draw(rightanglemark(A,W,C,5));dot("A", A, S);
dot("B", B, S);
dot("C", C, N);
dot("U", U, NE);
dot("V", V, NE);
dot("W", W, NW);
[/asy] geometryparallelogramright triangleisoscelesIsosceles Triangle