Let's position the vertex between the sides a and b at the origin and incline the 12-gon such that both sides make equal angles with the x-axis. The angle is 15 degrees (pi/12). Each next side has the angle of 30 degrees more.

The sum of all sides as vectors...

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Let's position the vertex between the sides a and b at the origin and incline the 12-gon such that both sides make equal angles with the x-axis. The angle is 15 degrees (pi/12). Each next side has the angle of 30 degrees more.

The sum of all sides as vectors is zero, consider the projection to the x-axis:

a cos15 + 2012 cos45 + 2013 cos75 + 2014 cos105 + 2015 cos135 + 2016 cos165 + 2017 cos195 + 2018 cos225 +2019 cos255 + 2020 cos285 + 2021 cos315 + b cos345 = 0 .

Now transform it slightly:

( a + b ) cos15 + ( 2012 - 2018 ) cos45 + ( 2013 - 2019 ) cos75 + ( 2014 - 2020 ) + ( 2015 - 2021 ) cos105 - ( 2016 + 2017 ) cos15 = 0 ,

( a+ b ) cos15 = 6 ( cos45 + cos75 + cos105 + cos135 ) + 4033 cos15,

a+b = 4033 + 0 = 4033.