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Heptagonal Triangle (Posted on 2019-12-27) Difficulty: 4 of 5
Let the vertices of △ABC be coinciding with the first, second and fourth vertices of a regular heptagon.

Find tan A + tan B + tan C.

No Solution Yet Submitted by Danish Ahmed Khan    
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soln - spoiler | Comment 2 of 4 |

All numbers below are approximate.

The interior angles are (180 deg -  360 deg /7) = 128.57 deg

There are four chords extending from C to vertices. They evenly split the interior angle into five angles of 25.71 deg apiece.  So C= 25.71 deg

Likewise angle B contains the sum of four such angles: So, B = 102.87 deg and A contains two: A= 51.42 deg

answer = tan(25.71) + tan(102.87) + tan(51.42)

I pause here due to technical difficulties.

(Why do I think there may be a more elegant solution?)

Edited on December 28, 2019, 9:34 am
  Posted by Steven Lord on 2019-12-28 09:31:51

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