Amiya Kumar 
📐 Concept: Medians and Centroid in a Triangle
In a triangle, the three medians (joining vertex to midpoint of opposite side) intersect at a point called the centroid, usually denoted as \( G \).
- Medians: \( AD, BE, CF \)
- Centroid \( G \) divides each median in the ratio \( 2:1 \)
That is,
\[
\frac{AG}{GD} = \frac{BG}{GE} = \frac{CG}{GF} = \frac{2}{1}
\]
💡 Properties:
- The centroid always lies inside the triangle.
- It is the center of gravity for a uniform triangular plate.
- Each median divides the triangle into two triangles of equal area.
🧪 CAT-Style Question on Centroid
Q. \( \triangle ABC \) is a triangle and \( G \) is its centroid. If \( AG = BG \), what is \( \angle BGC \)?
📝 Solution:
We know from centroid property that \( AG:GD = 2:1 \), but here we are told \( AG = BG \).
This implies triangle symmetry such that:
- \( AG = BG \)
- \( D \) is the midpoint of \( BC \)
- \( AG \perp BC \)
\[ \Rightarrow \angle BGC = 90^\circ \]
✅ Final Answer: \( \boxed{90^\circ} \)
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