MathDB

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]

Problem Statement