Prove Triangle Inequality: AB/MZ + AC/ME + BC/MD ≥ 2t/r

[solakis1]

Given a triangle ABC and a point M inside the triangle ,draw perpendiculars MZ,MD,ME at the sides AB,BC,AC respectively. Then prove:$$\frac{AB}{MZ}+\frac{AC}{ME}+\frac{BC}{MD}\geq\frac{2t}{r}$$

Where t is half the perimeter of the triangle and r is the radius of the inscribed circle

 

[solakis1]

[sp]Here also the Cauchy-Schwarz inequality may be used for the solution of the problem[/sp]