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: Understanding the Concept:
A quadrilateral is a parallelogram if its diagonals bisect each other.
We will use the properties of midpoints and given congruences to show the diagonals \(BC\) and \(OG\) bisect each other.
Step 2: Detailed Explanation:
In the quadrilateral \(OBGC\):
1. It is given that \(D\) is the mid-point of side \(BC\).
This implies that \(BD = DC\).
2. It is also given that \(AD\) is produced to \(G\) such that \(OD = DG\).
This implies that \(D\) is the mid-point of segment \(OG\).
3. In quadrilateral \(OBGC\), the diagonals are \(BC\) and \(OG\).
Since both diagonals intersect at \(D\) and are bisected at \(D\) (\(BD = DC\) and \(OD = DG\)), the diagonals bisect each other.
4. By the property of quadrilaterals, if the diagonals bisect each other, the quadrilateral is a parallelogram.
Therefore, \(OBGC\) is a parallelogram.
Step 3: Final Answer:
Since the diagonals \(BC\) and \(OG\) bisect each other at point \(D\), quadrilateral \(OBGC\) is a parallelogram.