Trigonometry & Inverse Trigonometry
Triangle Centers
Grade 11
Question:
<p>If the line joining the incentre to the centroid of a triangle ABC is parallel to the side BC. Which of the following are correct?</p>
<p>(a) 2b = a + c</p>
<p>(b) 2a = b + c</p>
<p>(c) \(\cot \frac{A}{2} \cot \frac{C}{2} = 3\)</p>
<p>(d) \(\cot \frac{B}{2} \cot \frac{C}{2} = 3\)</p>
Step-by-Step Solution
Key Concept: Use coordinate conditions for parallelism and relate incenter/centroid positions to side lengths
<p>The incentre I has coordinates proportional to (a, b, c) and centroid G has coordinates (a+b+c)/3 each.</p><p>For IG ∥ BC, the y-coordinates of I and G must have the same ratio as x-coordinates.</p><p>This leads to the condition: \(2a = b + c\)</p><p>From this, using half-angle formulas: \(\cot \frac{B}{2} \cot \frac{C}{2} = 3\)</p>
Correct Answer: B