4.5. THE SECTION FORMULA

To find the coordinates of a point which divides the join of the points (x ₁ , y ₁ , z ₁ ) and (x ₂ , y ₂ , z ₂ ) internally in the ratio m ₁ : m ₂ .

Let R( x, y, z) divide the join of given points P( x ₁, y ₁, z ₁) and Q( x ₂, y ₂, z ₂) in the ratio m ₁ : m ₂.

Draw PL, RN and QM ?s on the XY plane.

The lines PL, QM and RN clearly lie in one plane ( ? PL, QM, RN are ? to the same plane XY ; they are . Further these lines are cut by the same line PRQ. ? they are coplanar). Thus L, N, M, the feet of ?s lie in a straight line, the line of intersection of this plane and the plane XY.

Through R, draw a straight line ARB LNM, to meet LP (produced) in A and MQ in B.

Now in ?s APR and BRQ, we have

? The ?s APR and BRQ are similar, and so

Now

? From (1),

or

Similarly,

Hence the coordinates of the point which divides the join of (x ₁ , y ₁ , z ₁ ) and (x ₂ , y ₂ , z ₂ ),...