This was originally asked by Annie G without the condition of the triangle being acute-angled and gianlino had shown that the inequality does not hold for all triangles giving an example taking angles of the triangle as 30°, 30° and 120°. He informed me that the inequality is indeed true with the condition that the triangle is acute-angled, but I do not know the proof.
更新1:
I have extended time as I am out f station. Shall be back home by 10th and select BA after reading all answers.