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6.BW. DISTANCES TO CORNERS OF A SQUARE
New section. If ABCD is a rectangle, P is a point and a, b, c, d are the distances of P from the corners of the rectangle, then the basic relation a² + c² = b² + d² is easily shown. This leads to a number of problems. A little research has found references back to 1896, but the idea might be considerably older.
AMM 3 (1896) 155. ??NYS -- this is the earliest reference given by Trigg, cf below.

SSM 15 (1915) 632. ??NYS -- cited by Trigg, below.

AMM 35 (1928) 94. ??NYS -- cited by Trigg, below.

SSM 32 (1932) 788. ??NYS -- cited by Trigg, below.

NMM 12 (1937) 141. ??NYS -- cited by Trigg, below.

AMM 47 (1940) 396. ??NYS -- cited by Trigg, below.

NMM 16 (1941) 106. ??NYS -- cited by Trigg, below.

NMM 17 (1942) 39. ??NYS -- cited by Trigg, below.

AMM 50 (1943) 392. ??NYS -- cited by Trigg, below.

SSM 46 (1946) 89, 783. ??NYS -- cited by Trigg, below.

SSM 50 (1950) 324. ??NYS -- cited by Trigg, below.

SSM 59 (1959) 500. ??NYS -- cited by Trigg, below.

"A. Polter Geist", proposer; Joseph V. Michalowicz, Mannis Charosh, solvers, with historical note by Charles W. Trigg. Problem 865 -- Locating the barn. MM 46:2 (Mar 1973) 104 & 47:1 (Jan 1974) 56-59. For a square with an interior point, a, b, c = 13, 8, 5. How far is P from the nearest side? First solver determines the side, s, by applying the law of cosines to triangles BPA and BPC and using that angles ABP and BPC are complementary. This gives a fourth order equation which is a quadratic in s². Second solver uses a more geometric approach to determine s. The distances to the sides are then easily determined. Trigg gives 13 references to earlier versions of the problem -- see above.

Anonymous. Puzzle number 35 -- Eccentric lighting. Bull. Inst. Math. Appl. 14:4 (Apr 1978) 110 & 13:5/6 (May/Jun 1978) 155. Light bulb in a room, with distances measured from the corners. a, b, c = 9, 6, 2. Find d. Solution uses the theorem of Apollonius to obtain the basic equation.

David Singmaster, proposer and solver. Puzzle number 40 -- In the beginning was the light. Bull. Inst. Math. Appl. 14:11/12 (Nov/Dec 1978) 281 & 15:1 (Jan 1979) 28. Assuming P is inside the rectangle, what are the conditions on a, b, c for there to be a rectangle with these distances? When is the rectangle unique? When can P be on a diagonal? Solution first obtains the basic relation, which does not depend on P being in the rectangle. Reordering the vertices if necessary, assume b is the greatest of the distances. Then a² + c²  b² is necessary and sufficient for a rectangle to exist with these distances. This is unique if and only if equality holds, when P = D. If the distances are all equal, then P is at the centre of the rectangle, which can have a range of sizes. If the distances are not all equal, there is a unique rectangle having P on a diagonal and it is on the diagonal containing the largest and smallest distances.

Marion Walter. Exploring a rectangle problem. MM 54:3 (1981) 131-134. Takes P inside the rectangle with a, b, c = 3, 4, 5. Finds the basic relation, noting P can be anywhere, and determines d. Then observes that the basic relation holds even if P is not in the plane of ABCD. Ivan Niven pointed out that the problem extends to a rectangular box. Mentions the possibility of using other metrics.

James S. Robertson. Problem 1147 -- Re-exploring a rectangle problem. MM 55:3 (May 1982) 177 & 56:3 (May 1983) 180-181. With P inside the rectangle and a, b, c given, what is the largest rectangle that can occur? Observes that a² + c² > b² is necessary and sufficient for a rectangle to exist with P interior to it. He then gives a geometric argument which seems to have a gap in it and finds the maximal area is ac + bd.

I. D. Berg, R. L. Bishop & H. G. Diamond, proposers. Problem E 3208. AMM 94:5 (May 1987) 456-457. Given a, b, c, d satisfying the basic relation, show that a rectangle containing P can have any area from zero up to some maximum value and determine this maximum.

Problem 168.2. M500 168 ??? Seven solvers, M500 170 (Oct 1999) 15-20. Given P inside a square and a, b, c = 5, 3, 4, find the side of the square.