Prove that, if (c-b).a = 0 and (c-a).b = 0, then (b-a).c=0. Show that this can be used to prove the following geometrical results.
(a) The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point.
I can do the first part by using the distributive rule and communitative rule for scalar products.
But part (a) took a too much effort for what I think is meant to be a simple question. I showed that the lines through the vertices can be expressed in the one vector, therefore they must meet in a point. I used the fact that their dot product is zero. It took a hell of a lot of tedious algebra and I'm not even sure that its right. I'd appreciate any help.