Question

D is the mid-point of side BC of \(\triangle ABC\). CE and BF intersect at O, a point on AD. AD is produced to G such that \(OD = DG\). Prove that OBGC is a parallelogram.

Hint:

To prove a quadrilateral is a parallelogram, diagonal bisection is often the fastest method when midpoints are given.

Answer 1

Step 1: Identifying the Quadrilateral:
We consider quadrilateral OBGC.
To prove it is a parallelogram, we use the property:
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Step 2: Using Given Midpoint Information:
Given D is the midpoint of BC.
Therefore,
BD = DC

Also, AD is produced to G such that:
OD = DG

Thus, D is the midpoint of OG.

Step 3: Identifying the Diagonals:
In quadrilateral OBGC,
Diagonal 1 = BC
Diagonal 2 = OG

These two diagonals intersect at point D.

Since,
BD = DC
OD = DG

Both diagonals are bisected at D.

Step 4: Applying Parallelogram Property:
A quadrilateral whose diagonals bisect each other is a parallelogram.

Therefore,
OBGC is a parallelogram.

Final Answer:
Since diagonals BC and OG bisect each other at D, quadrilateral OBGC is a parallelogram.