Question

perpendiculars are drawn from a point within an equilateral triangle to the three sides. prove the sum of the three perpendiculars is equal to that of the altitude of the triangle.​

Answer 1

Please find the attachment and refer to the explanation below.

As given in the question, perpendiculars OD, OE and OF are drawn from 0 to sides AB, BC and AC respectively of the equilateral triangle ABC

Area of triangle OBC = (1/2)*BC*OE            (1)

Area of triangle OAC = (1/2)*AC*OF            (2)

Area of triangle AOB = (1/2)*AB*OD           (3)

Let sides of ABC be a

using (1), (2) and (3),

Area of ABC = Area of OBC +Area of AOB +Area of AOB =                      (1/2)*a*( OD+OE+OF)                                   (4)

if h is the altitude of AABC, then area of ABC = (1/2)*a*h         (5)

hence from (4) and (5),

(1/2)*a*( OD+OE+OF)  =  (1/2)*a*h

⇒ OD+OE+OF = h

So, we can say sum of perpendiculars (OD+OE+OF) is equal to altitude h