Geometry Problem

Simon YAU

2014-01-12 1:09 pm

Let P be any point on a plane, for any convex or concave polygon on the plane, reflect P about the vertexes of the polygon to obtain a new set of points, prove that the new polygon formed by joining the set of new points is similar to the original polygon. Further, is this result true for figure of any shape? Give a proof.
Can anyone help?
Thanks.

回答 (1)

少年時

2014-01-12 5:10 pm

✔ 最佳答案

Yes.
Assume point P inside the polygon a1, a2, a3, ..., an, the reflected points b1, b2, b3, ..., bn form a new polygon.
As Pa1= a1b1, Pa2 = a2b2, Pa3 = a3b3, ..., Pan= anbn, by using mid-pt theorem, a1b1 // a2b2, a2b2 // a3b3, ..., ana1 // bnb1.
Also, b1b2 = 2a1a2, b2b3 = 2a2a3, ..., bnb1 = 2ana1.
Since a1a2/b1b2 = a2a3/b2b3 = ... = ana1/bnb1 = 1/2,
and angle Pa1a2 = angle Pb1b2, angle Pa2a3 = angle Pb2b3, ...
so polygon a1a2a3...an is similar to polygon b1b2b3...bn (all sides with same ratio and all corresponding angles are equal)).

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