It’s easy to show that the five acute angles in the vertices
of a
regular 5pointed star total 180°.
Please show that the sum of these angles in an irregular 5pointed star is also 180°.
Source: A. Korshkov, the Russian science magazine Kvant.
A 'regular 5pointed 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 5pointed 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 60degree angles in the vertices, is just one example of such a star.
Edited on December 14, 2016, 10:49 pm

Posted by broll
on 20161214 22:35:32 