26th Eötvös 1922 Problem 1 Show that given four non-coplanar points A, B, P, Q there is a plane with A, B on one side and P, Q on the other, and with all the points the same distance from the plane.
Solution
Let U be the midpoint of AQ, V the midpoint of AP, W the midpoint of BP, and X the midpoint of BQ. Then UV and WX are both parallel to PQ and hence to each other. Similarly, XU and VW. So UVWX is a parallelogram and hence lies in a plane. This plane is the one required. AB is parallel to the plane, so A and B are equidistant from it. W is the midpoint of PB, so P and B are equidistant from the plane. Similarly, X is the midpoint of BQ, so B and Q are equidistant from the plane. So all four points are equidistant. 26th Eötvös 1922 © John Scholes [email protected] 1 Nov 2003 Last corrected/updated 1 Nov 03
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