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the midpoint of
The locus point is the perpendicular bisector of AB. The locus point is the perpendicular bisector of AB.
The Law of Cosines: c^2=a^2 + b^2 -2abcos(ab) , c is the distance between the two points a and b and (ab) is the angle between a and b from the origin. If one point is taken as the origin, and a and b a re taken at right angles to each other, then cos(ab) is zero and you have Pythagora' Theorem..
The answer depends on the information that you have about the four points and the manner in which that information is presented. Suppose the 4 points are A, B, C and D and the point that you find is P. If you have the coordinates of A, B, C and D then gradient AP = gradient AB (or any other pair) will suffice. If you have any one of vectors AB (or AC, AD, BC, BD), then vector AP is parallel to vector AB will suffice.
an example would be if you had line AB who's points are (5,3) and another at (1,5). Connecting diagonally. If you were to put a vertical line anywhere between the points. it only goes through one point. Making it a function.