If D,E and F are the mid pts of sides AB, BC and CA respectively of an equilateral triangle ABC, prove that triangle DEF is itself an equilateral triangle.
MID PTS THEOREM
ΔABC is equilateral triangle, A = B = C = 60
and AB = BC = AC,
AD = DB.
AD + DB = AB
2 AD = AB
AD = AB/2
Similarly AF = AC/2
AF = AD since AC = AB.............(2)
A = 60 and AFD = ADF .....from (2)
Hence AFD = ADF = 60.
BD = BE
B = 60 and BDE = DEB .....from (2)
Hence BDE = DEB = 60.
BDE + ADF + FDE = 180.
FDE = 60.
Similarly in ΔDEF, the remaining two angles can be shown to be 60.
Hence triangle DEF is an equilateral triangle.