20 sides polygon

Let $ [A_1A_2A_3 \ldots A_{20}] $ be a polygon of 20 sides, in the plane. We represent by $ \measuredangle A_i $ the measure of the internal angle of the polygon in the vertex $ A_i $ for $ i = 1, 2, 3, \ldots , 20 $. Supose that the polygon has the following properties:

Every side has the same length;
$\measuredangle A_1 = \measuredangle A_3 = \measuredangle A_5 = \cdots = \measuredangle A_{19} = 216^\circ$ ;
$\measuredangle A_2 = \measuredangle A_4 = \measuredangle A_6 = \cdots = \measuredangle A_{20} = 108^\circ$

Show that the lines $A_1A_{11}$ , $A_2A_{14}$ , $A_3A_{17}$ , $A_5A_{19}$ and $A_8A_{20}$ have a point in common.

#Geometry

Easy Math Editor

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Comments

What have you tried?

Calvin Lin Staff - 4 years, 5 months ago

@Calvin Lin I tried dividing it into two pentagons, but then I wasn't able to do much so I thought about setting up an axis and work with point coordinates and vectors, but it was also a dead-end. Maybe there's some property or something I'm missing. What do you suggest?