Inequality in triangles
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A fan is composed of a set of similar isosceles triangles arranged side by side so that apart from the end triangles each triangle shares an equal side with its neighbours. The triangles are allowed to fold into one another face to face or back to back so the the fan can be closed up. When the fan is fully opened the bases of the triangles form the curved edge of the fan, but this "curve" is actually made up of straight edges. When the fan is half open the triangles are partly folded and the curved triangle of the fan has a smaller angle than when the fan is fully opened. So we can say that the angle of the fan is less than or equal to the combined angles of the isosceles triangles. We can also say that the apparent length of the fan "curve" is less than or equal to the combined bases of the triangles. And we can say that the more triangles there are the more the curve of the fan approximates to an arc of a circle, but it will always be less than the length of such an arc, because the number of triangles can never be infinite. If the angle of the fan when extended is x radians then the length of a true arc is rx, where r is the common height of the triangles, the length of the fan or the radius of the arc. If b is the common length of the triangle base and there are n triangles, nb<rx.

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