Question

G is the concarance of median of triangle DEF.Take point H on ray DG such that D-G-H and DG=GH.Then prove that rectangle GEFH is a parallelogram

Answer 1

In the triangle ABC draw medians BE, and CF, meeting at point G. Construct a line from A through G, such that it intersects BC at point D.

We are required to prove that D bisects BC, therefore AD is a median, hence medians are concurrent at G (the centroid).

Proof:

Produce AD to a point P below triangle ABC, such that AG = GP.

Construct lines BP and PC.

Since AF = FB, and AG = GP, FG is parallel to BP. (Euclid)

Similarly, since AE = EC, and AG = GP, GE is parallel to PC

Thus BPCG is a parallelogram.

Since diagonals of a parallelogram bisect one another (Euclid), therefore BD = DC.

Thus AD is a median. QED

Corollary: GD = AD/3.

Proof:

Since AG = GP and GD = GP/2, AG = 2GD.

AD = (AG + GD) = (2GD + GD) = 3GD.

Hence GD = AD/3. QED