James

A pencil of lines is just a set of lines that all meet at a certain point called the centre of the pencil.

if 3 lines L1,L2 and L3 are lines of the same pencill then there exist numbers k1,k2 and k3 such that k1xL1+k2xL2+k3xL3 =0

in our case k1=k2=k3=1 ie. if you just add all the three equations of the lines I gave you they will be =0 this means that the three lines are members of the same pencil

in other words they meet in the same point all of them this point is the well known point called the barycenter of the trangle ABC and its coordinates are the ones I mntioned above.

Now go here and check what i mean by determinant.

Determinant - Wikipedia, the free encyclopedia
if 3 lines meet at a point their coefficients form a 3x3 determinant that is = 0

sent me a private message if you need more explanations.

Minoas