MathDB

Consider the two triangles

ABC

and

PQR

shown below. In triangle ABC, \angle ADB \equal{} \angle BDC \equal{} \angle CDA \equal{} 120^\circ. Prove that x\equal{}u\plus{}v\plus{}w. [asy]unitsize(7mm); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair C=(0,0), B=4*dir(5); pair A=intersectionpoints(Circle(C,5), Circle(B,6))[0];
pair Oc=scale(sqrt(3)/3)*rotate(30)*(B-A)+A; pair Ob=scale(sqrt(3)/3)*rotate(30)*(A-C)+C;
pair D=intersectionpoints(Circle(Ob,length(Ob-C)), Circle(Oc,length(Oc-B)))[1];
real s=length(A-D)+length(B-D)+length(C-D); pair P=(6,0), Q=P+(s,0), R=rotate(60)*(s,0)+P; pair M=intersectionpoints(Circle(P,length(B-C)), Circle(Q,length(A-C)))[0];
draw(A--B--C--A--D--B); draw(D--C);
label("

B

",B,SE); label("

C

",C,SW); label("

A

",A,N); label("

D

",D,NE); label("

a

",midpoint(B--C),S); label("

b

",midpoint(A--C),WNW); label("

c

",midpoint(A--B),NE); label("

u

",midpoint(A--D),E); label("

v

",midpoint(B--D),N); label("

w

",midpoint(C--D),NNW);
draw(P--Q--R--P--M--Q); draw(M--R);
label("

P

",P,SW); label("

Q

",Q,SE); label("

R

",R,N); label("

M

",M,NW); label("

x

",midpoint(P--R),NW); label("

x

",midpoint(P--Q),S); label("

x

",midpoint(Q--R),NE); label("

c

",midpoint(R--M),ESE); label("

a

",midpoint(P--M),NW); label("

b

",midpoint(Q--M),NE);[/asy]

Problem Statement