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Problem 177

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Integer angled Quadrilaterals

Let ABCD be a convex quadrilateral, with diagonals AC and BD. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles.

For example, at vertex A, the two angles are CAD, CAB.

We call such a quadrilateral for which all eight corner angles have integer values when measured in degrees an “integer angled quadrilateral”. An example of an integer angled quadrilateral is a square, where all eight corner angles are 45°. Another example is given by DAC = 20°, BAC = 60°, ABD = 50°, CBD = 30°, BCA = 40°, DCA = 30°, CDB = 80°, ADB = 50°.

What is the total number of non-similar integer angled quadrilaterals?

Note: In your calculations you may assume that a calculated angle is integral if it is within a tolerance of 10^-9 of an integer value.

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整数角度四边形

ABCD是一个凸四边形，对角线为AC和BD。在四边形的每个顶点，对角线和相邻的两条边各构成一个角，总共构成八个这样的角。

例如，在顶点A的两个角分别为角CAD和角CAB。

当一个四边形的所有八个角均为整数角度时，我们称其为“整数角度四边形”。一个整数角度四边形的例子是正方形，它的八个角都是45°。另一个例子是DAC = 20°, BAC = 60°, ABD = 50°, CBD = 30°, BCA = 40°, DCA = 30°, CDB = 80°, ADB = 50°。

不考虑相似，整数角度四边形的总数是多少？

注意：在你的计算中，你可以假定一个角的的角度为整数，如果它与一个整数值的误差在10^-9以内。

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