How to Construct the Incircle of a Triangle

The incircle of a triangle is the largest circle that fits inside it, touching all three sides. Its center is the incenter, found where the three angle bisectors meet.

• Straightedge or Ruler: For creating straight lines and measurements.
• Compass: Indispensable for constructing arcs and circles.
• Pencil: For marking and drawing.

Worked examples: constructing the incircle

Practice questions on the incircle of a triangle

1. Can an obtuse triangle have an incircle that touches all its sides? Explain.
2. Given an equilateral triangle of side length \(9 \text{ cm}\), how would you find its inradius without constructing the incircle?
3. Does the incenter of a triangle always lie within the triangle?

1. Yes, every triangle, including obtuse triangles, has a unique incircle that touches all its sides.
2. For an equilateral triangle with side length \(a\), the inradius \(r\) can be found using the formula:
   \( r = \frac{a}{2\sqrt{3}} \)
   For \(a = 9 \text{ cm}\), \(r = \frac{9}{2\sqrt{3}} = \frac{9\sqrt{3}}{6} = 1.5\sqrt{3} \text{ cm}\).
3. Yes, the incenter always lies inside the triangle.

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