Problem

Source: Czech-Polish-Slovak Junior Match 2015, Individual p1 CPSJ

Tags: geometry, max, area of a triangle, angle, right triangle



In the right triangle $ABC$ with shorter side $AC$ the hypotenuse $AB$ has length $12$. Denote $T$ its centroid and $D$ the feet of altitude from the vertex $C$. Determine the size of its inner angle at the vertex $B$ for which the triangle $DTC$ has the greatest possible area.