1247. When the sum equals the product

    In a certain exercise the sum of the tangents of the angles of a triangle was given, and the product of these three tangents was requested... Many students said they needed more data but Pepe Chapuzas calculated it.

    Dear Teacher:

    The sum equals the product...

tg α + tg β + tg γ  =

=  tg α + tg β + tg (180º−α−β)  =

=  tg α + tg β – tg (α+β)  =

=  tg α + tg β − (tg α + tg β) / (1 − tg α · tg β)  =

=  [ tg α + tg β – (tg α + tg β) · tg α · tg β − tg α − tg β ] / (1 − tg α · tg β)  =

=  − (tg α + tg β) · tg α · tg β / (1 − tg α · tg β)  =

=  − tg α · tg β · tg (α+β)  =

=  tg α · tg β · tg (180º−α −β)  =

=  tg α · tg β · tg γ

... but not always, because the right angles...