If D, E and F be the middle points of the sides BC,CA and AB of the `DeltaABC`, then `AD+BE+CF` is

Options

A. centroid of the triangle DEF is the same as that of ABC
B. orthocenter of the triangle DEF is the circumcentre of ABC
C. orthocenter of the triangle DEF is the incenter of ABC
D. centroid of the triangle DEF is not the same as that of ABC

Answer

A, B

Explanation

Let D, E and F be the mid-point of the sides of `DeltaABC`.
E and F are mid-point of AC and AB, respectively

So, `EF||BC and EF = (1)/(2)BC`
Thus, AD will also divide EF into equal parts.
Hence, DH is also median of `DeltaDEF`
Similarly BE, median of `DeltaABC`, is also median of `DeltaDEF`.
These two lines meet at G.So the centroids of both triangles are same.
The orthocenter, O of `DeltaDEF` is the point of intersection of the perpendiculars DH and EK drawn from D and E, respectively.

Since, `EF||BC`, DH is perpendicular to BC also.
Similarly EK is perpendicular to AC.
So, orthocenter of `DeltaDEF` and circumcentre of `DeltaABC` is the same points