Question

in triangle abc d, e, f are midpoints of side ab, bc, ac respectively p is the foot of the perpendicular of a to side bc. show that points d,e,f,p are concyclic

Answer 1

Given ∆ABC is an equilateral triangle and D , E ans F are mid-points of BC , AC and AB respectively.

TO PROVE : ∆FED is an equilateral triangle.

Proof :
Since D and E are mid-points of BC and AC respectively.
DE = 1 / 2 AB ………...(i)
[By mid point theorem ,the line segment joining the mid-points of two sides of a triangle is half of the third side. ]

Similarly ,E and F are the mid - points of AC and AB respectively .

∴ EF = 1 / 2 BC ……….(ii)

F and D are the mid - points of AB and BC respectively .

∴ FD = 1 / 2 AC ………...(iii)

Now, △ABC is an equilateral triangle .
AB = BC = CA
1/2 AB = 1/ 2 BC = 1/ 2 CA
DE = EF = FD
[From eq (i) , (ii) , (iii) ]

Hence, ∆FED is an equilateral triangle .


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