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Question

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.

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Solution



From P, Q and R, draw perpendicular PL, QM and RN on the xy-plane. Also, draw PSQM, meeting QM and RN at S and T respectively. From similar triangles PRT and PQS, we have

RTQS=PRPQ

RNTNQMSM=mm+n

RNPLQMPL=mm+n

zz1z2z1=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.


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