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.

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**