Consider any three vectors u,v,w in 3-dimensional space s.t joining the three vectors by straight lines forms a triangle. Under what condition on c,d,e will the combination cu + dv + ew fill the triangle?
This is a challenge question from Gilbert Strang's book. The answer is c>=0,d>=0, e>=0 and c + d + e = 1. All the special points of the triangle (1,0,0),(0,1,0),(0,0,1),(.5,.5,0),(0,.5,.5),(.5,0,.5),(1/3,1/3,1/3) meet this criteria. But how can i prove that all the points on and inside the triangle obey this condition.
You can place $u,v,w$ to $(0,0,0), (0,1,0), (0,0,1)$ via a linear isomorphism. For such triple it is immediate to check the property