Problem:
Four distinct points are arranged in a plane so that the segments connecting them have lengths , and . What is the ratio of to ?
Answer Choices:
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Solution:
Since of the segments have length , some of the points (call them , and ) must form an equilateral triangle of side length . The fourth point must be a distance from one of , or , and without loss of generality it can be assumed to be . Thus lies on a circle of radius centered at . The distance from to one of the other points (which can be assumed to be ) is , so is a diameter of this circle and therefore is the hypotenuse of right triangle with legs of lengths and . Thus , and the ratio of to is .
The problems on this page are the property of the MAA's American Mathematics Competitions