Question

in an equilateral triangle o is any point in interior of the triangle and perpendicular are drawn from o to the sides prove that sum of perpendicular is constant

Answer 1

Please see the attached picture of triangle.
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Draw segment connecting O with each if its vertices.
now we get three triangles, ΔAOB,ΔAOC and  ΔBOC.
OD, OE and OF are perpendicular to sides of ΔABC, so they are the heights of their respective triangles.

Let height of ΔABC be 'h'
so we can write
area (ΔAOB )+area (ΔAOC)+area (ΔBOC)=area(ΔABC)

we know all the sides of the triangle are equal  say 'a'.
using height formula for area of triangle
we have,
(a.OD)/2+a.OE)/2+(a.OF)/2=(h.a)/2
divide both sides by (a/2)
we get, OD+OE+OF=h

Hence proved that the sum of perpendicular to the sides from a point is constant, because for a triangle the height is constant.