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 It's all in the stars (Posted on 2016-12-14)
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 No Rating

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 Solution | Comment 1 of 2

Let A, B, C, D, and E label (in that order) the five points
of the star.

Let I = AC /\ BE,
J = BD /\ CA,
K = CE /\ DB,      /\  denotes intersection
L = DA /\ EC,

5*180 is the sum (in degrees) of the interior angles of the
following five triangles: CIE, DJA, EKB, ALC, and
BMD.

3*180 is the sum (in degrees) of the interior angles of the
pentagon IJKLM.

Star is the sum (in degrees) of angles A, B, C, D, and E.

Therefore,

2*Star + 3*180 = 5*180

or

Star = 180 degrees.

QED

 Posted by Bractals on 2016-12-14 13:15:45

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