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It's all in the stars (Posted on 2016-12-14) Difficulty: 2 of 5
It’s easy to show that the five acute angles in the vertices
of a regular 5-pointed star total 180°.

Please show that the sum of these angles in an irregular 5-pointed star is also 180°.

Source: A. Korshkov, the Russian science magazine Kvant.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts A counterexample. Comment 2 of 2 |

A 'regular 5-pointed star' could be either a regular star pentagon (pentagram) or a regular concave decagon. The above proposition is true of the first but not necessarily of the second.

As a seasonal sketchpad pastime:

Construct regular pentagon ABCDE, with C pointing upwards, and the perpendiculars A to CD at L, B to DE.

P is a point on AL closer to L than A. Construct a line through P parallel to AB, crossing DE at Q. Construct inverted regular pentagon PQRST.

Now the polygon ASBTCPDQER is a regular 5-pointed star and the angles in its vertices can be anything from just over 0 to just less than 108 degrees.

The Mario Brothers Super Star, with 60-degree angles in the vertices, is just one example of such a star.


Edited on December 14, 2016, 10:49 pm
  Posted by broll on 2016-12-14 22:35:32

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