Problem:
A point is to be chosen in the coordinate plane so that it is equally distant from the -axis, the -axis, and the line . Then is
Answer Choices:
A.
B.
C.
D.
E. not uniquely determined
Solution:
From a picture we can determine that there must be four such points, with different coordinates. To be equidistant from two nonparallel lines and , a point must be on either of the two lines which bisect a pair of opposite angles at the intersection of and . In particular, to be equidistant from the and axes, must be on either or (shown dashed in the adjoining figure); to be equidistant from the -axis and must be on one of the lines through shown dotted in the figure. There are four points simultaneously on one of the dashed and one of the dotted lines, as indicated. One can show that the coordinates of the four points, in the order labeled, are and .
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The line and the axes determine a triangle. This triangle has an inscribed circle and three escribed circles. The centers of these circles, and no other points, satisfy the equal-distance condition. Hence there are four points. It is easy to see by a sketch that their -coordinates are not all the same.