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Problem 577

AMC 12 late, AIME early

Geometry Difficulty 5.0 Find the answer

Let's construct a symmetric trapezoid, given the length of the non-parallel sides $(c)$ , the length of the diagonal $(d)$ , and the angle between the diagonals!

Let's examine the possibility of construction, especially in the case when the angle opposite the non-parallel sides is $60^{\circ}$ .

Your answer

Official solution

I. Solution. Let $ABCD$ be the sought trapezoid and $ABc$ , so that the trapezoid is convex.

If $\alpha=60^{\circ}$ , then $c$ is equal to the side of the regular hexagon inscribed in the circle; the diameter of the circle is $2c$ and the condition for the possibility of the construction is

$c<d<2c$

the same as under I.

[^0]: ${ }^{1}$ If it is a convex trapezoid, then $c<d$ !

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.