Let P ( x 1 y 1 z 1 ) and Q ( x 2 y 2 z 2 ) be the two given points. Let R(x,y,z) divide [PQ] in the given ratio m 1 : m 2 such that

Unknown control sequence 'fracPRRQ ' .

Draw PM, QN and RL perpendiculars to the XY-plane. Through R, draw a straight line ARB parallel to MLN to meet MP (produced) and NQ in points A and B respectively.

[Since PM, QN, RL are perpendiculars to the XY-plane, are parallel,and as they are cut by the line PRQ, they are coplanar. Points M,L, N lie on a straight line which is the intersection of this plane with XY-plane. Therefore, a line through R parallel to MLN also lies in that plane and hence meets MP (produced) and NQ.]

Clearly, s APR and RBQ are similar,

Unknown control sequence 'fracPABQ ' ..............(i)

From figure,

PA = M A − M P = L R − M P = z − z 1 and

BQ = N Q − N B = N Q − L R = z 2 − z .

From (i), we get

Unknown control sequence 'fracz '

( m 1 + m 2 ) z = m 1 z 2 + m 2 z