AD,BE and CF,the medians of triangle ABC are equal.then prove that triangle ABC is a equilateral triangle.

Asked by abinash.gupta003 | 26th Aug, 2016, 03:46: PM

Expert Answer:

begin mathsize 14px style Let space the space emdians space intersect space at space point space straight M.
rightwards double arrow straight M space is space centroid space of space triangle space ABC.
Now comma space centroid space divides space the space median space in space teh space ratio space 2 space colon space 1
Thus comma space we space have
AM space colon space MD space equals BM space colon space ME space equals CM space colon space MF space equals space 2 space colon space 1 space space
The space question space says space that space the space medians space are space equal space in space length. space
So space let space us space say space
AM space equals space BM space equals space CM space equals space 2 space units space and space MD space equals space ME space equals space MF space equals space 1 space unit
Now space consider space triangle CDM space and space triangle AFM comma
CM space equals space AM
FM space equals space MD
angle AMF space equals space angle CMD
rightwards double arrow triangle CDM space approximately equal to triangle CMD
rightwards double arrow AF space equals space CD
But space straight F space is space the space mid minus point space of space AB comma space
rightwards double arrow AF space equals space FB
rightwards double arrow AF space equals space FB space equals space BD space equals space CD
Hence comma space AF space plus space FB space equals space BD space plus space CD
rightwards double arrow AB space equals space BC
Similarly comma space we space can space show space BC space equals space AC space and space space hence space AB space equals space BC space equals space AC
rightwards double arrow ABC space is space an space equilateral space triangle. end style

Answered by Rashmi Khot | 26th Aug, 2016, 05:44: PM

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