This problem uses notions and results from Problems2.6 and 2.7. In view of (a) and (b) of Problem2.7, it is natural to extend the notion of barycentric coordinates of appoint in A2 as follows. Given any affine frame(a,b,c) in A2,we will say that the bary centric coordinates (x,y,z) of a point M, where x+y+z=1, are the normalized bary centric coordinates of M. Then, anytriple (x,y,z) such that x+y+z = 0 is also called as y stem of bary centric coordinates for the point of normalized bary centric coordinates
With this convention, the intersection of the two lines D and D′ is either a point or a vector, in both cases of bary centric coordinates
When the above is a vector, we can think of it as a point at infinity (in the direction of the line defined by that vector).
Let(D0,D′0),(D1,D′1),and(D2,D′2) be three pairs of six distinct lines, such that the four lines belonging to any union of two of the above pairs are neither parallel nor con current
(have a common intersection point). If D0 and D′0 have a unique intersection point, let M be this point, and if D0 and D′0 are parallel, let M denote anonnull vector defining the common direction of D0 and D′0. In either case, let (m,m′,m′′) be the bary centric coordinates of M, a sex plained at the beginning of the problem. We call M the intersection of D0 and D′0. Similarly, define N=(n,n′,n′′) as the intersection of D1 and D′1, and P=(p,p′,p′′) as the intersection of D2 and D′2.
Prove that
Iff either
(i) (D0,D′0),(D1,D′1),and(D2,D′2)are pairs of parallel lines; or
(ii)the lines of some pair (Di,D′i) are parallel, each pair (Dj,D′j) (with j=i) has a unique intersection point, and the set w o inter section points are distinct and determine a line parallel to the lines of the pair (Di,D′i);or
(iii)each pair (Di,D′i) (i=0,1,2)has a unique intersection point, and these points M,N, P are distinct and collinear.