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Perfect Beauty of math in the Symmetry


Symmetry -01

Look at the above figures. Which of them are symmetric? Which of them are not?

For the symmetric figures, what are the lines of symmetry (if applicable)?
In a symmetric figure, we can produce as many isosceles as we like. Do you know the easiest way to find them?

We need to explain on the last figure (the figure at the bottom right). It is consisted of two circular arcs with different radii. The two arcs are glued smoothly (at the gluing point). This last figure is not symmetric.

Not only this figure is NOT symmetric, but also we CANNOT pick out from it a sub-figure that is symmetric, as long as the figure contains both arcs with different radii. But can you argue for this claim?

最后一个图形是非对称的。 不仅全图非对称,而且你找不到对称的子图 (只要子图中包含了半径不同的圆弧段)!

Sometimes a figure might not be perfect under the given condition, but we are tempted to produce a figure that is more perfect than the one given in the question. For example, we might draw an isosceles when the given one is scalene, and draw a special quadrilateral (e.g. a rectangle) when the given one is just a trapezoid, or even arbitrary quadrilateral. This is the Pitfall! that we shall avoid.

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