Answer to Question #95173 in Geometry for Mashud

Question

Let A, B, C, D be four points in space. If P,Q,R,S are points lying on AB, BC, CD, DA respectively such that:
AP/PB=λ , BQ/QC= u, CR/RD= γ, DS/SA=δ.
Prove that P, Q, R, S are coplanar provided
λ u γ δ=1

Answer 1

In the problem it is necessary to prove that if lambda=1, u=1, gamma=1, delta = 1, then points P, Q, R and S are coplanar.

if lambda=1, u=1, gamma=1, delta = 1, then AP = PB, BQ = QC, CR = RD and DS = SA

Consider triangles ACB and ACD

if AP=PB and CQ=QB then PQ is the mid line of the triangle ACB,

then PQ=AC/2 and PQ || AC

if AS=SD and DR = CR then SR is the mid line of triangle ACD,

then SR = AC/2 and SR || AC

if SR || AC and PQ || AC, then SR || PQ (*)

if PQ=AC/2 and SR = AC/2 then PQ=AC (**)

Similarly SP = RQ and SP || RQ (***) (if triangles ADB and CDB are considered)

if (*) and (**) and (***) are all true, then SRPQ is a parallelogram.

if SRPQ is a parallelogram then vectors SR, SP, SQ lie in one plane, these vectors are coplanar and points S, R, P, Q are coplanar too.