Let P(x1, y1, z1) and Q(x2, y2, z2) be two points and let R(x, y, z) be a point on PQ dividing it in the ratio m:n. Prove that
x=mx2+nx1m+n, y=my2+ny1m+n and z=mz2+nz1m+n.
From P, Q and R, draw perpendicular PL, QM and RN on the xy-plane. Also, draw PS⊥QM, meeting QM and RN at S and T respectively. From similar triangles PRT and PQS, we have
RTQS=PRPQ
⇒RN−TNQM−SM=mm+n
⇒RN−PLQM−PL=mm+n
⇒z−z1z2−z1=mm+n
⇒z=mz2+nz1m+n
Similarly, x=mx2+nx1m+n and y=my2+ny1m+n
Hence, x=mx2+nx1m+n, y=my2+ny1m+n, mz2+nz1m+n.