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imc

geometry intermediate

Problem

For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral? a square a rectangle that is not a square a rhombus that is not a square a parallelogram that is not a rectangle or a rhombus * an isosceles trapezoid that is not a parallelogram
(A)
(B)
(C)
(D)
Solution
This question is simply asking how many of the listed quadrilaterals are cyclic (since the point equidistant from all four vertices would be the center of the circumscribed circle). A square, a rectangle, and an isosceles trapezoid (that isn't a parallelogram) are all cyclic, and the other two are not. Thus, the answer is .
Final answer
C