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How can he prove the points $A=(a_1, a_2, a_3)$, $B=(b_1, b_2, b_3)$, $C=(c_1, c_2, c_3)$, $D=(d_1, d_2, d_3)$ belong to the same plane, as if they belong can then find plane. I know how to prove three given points belong the same plane.

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2 Answers

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Hint: Translate by $-A$ so that we can assume $A=(0,0,0)$. Then points $A,B,C,D$ lie on a plane iff vectors $B,C,D$ do not span whole space, i.e. if $$\left| \begin{array}{ccc} b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{array} \right|=0$$

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Take any three of the points and determine the equation of the plane. As TonyK said, three points always belong to one plane and, if they do not all lie in a line, then the determine a unique plane. Once you have the equation of the plane, put the coordinates of the fourth point into the equation to see if it is satisfied. If the three points you chose do happen to lie on a single line then you are done- any fourth point will determine a plane that all four points lie on.

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