Show that the sum of the three altitudes of a triangle is less than the sum of three sides of the triangle. GIVEN : A ` A B C` in which `A D_|_B C ,B E_|_A C` and `C F_|_A Bdot` PROVE : `A D+B E+C F

Explanation

Let ABC be the triangle with sides BC, AC, AB opposite angles A, B, C being in length equal to a, b, c respectively.
Draw perpendiculars AD, BE and CF from A, B, C to opposite sides meeting sides BC, AC and AB at D, E, F respectively.
Now perpendicular `AD^2 = AC^2 - CD^2 => AD^2 < AC^2 or AD < AC or AD < b` -----(1)
Also, `BE^2 = AB^2 - AE^2 => BE^2 < AB^2 or BE < AB or BE < c` ------(2)
Likewise , `CF^2 = BC^2 - BF^2 => CF^2 < BC^2 or CF < BC or CF < a` ----(3)
Adding inequalities (1), (2), (3),` AD + BE + CF < A B +B C+ A C`