Problem:
Three noncollinear points and a line are given in the plane. Suppose no two of the points lie on a line parallel to (or itself). There are exactly lines perpendicular to with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of .
Solution
Solution:
The condition for the line is that each of the three points lies at an equal distance from the line as from some fixed point; in other words, the line is the directrix of a parabola containing the three points. Three noncollinear points in the coordinate plane determine a quadratic polynomial in unless two of the points have the same -coordinate. Therefore, given the direction of the directrix, three noncollinear points determine a parabola, unless two of the points lie on a line perpendicular to the directrix. This case is ruled out by the given condition, so the answer is 1.
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