Key takeaway: The incenter is the point where the three angle bisectors of a triangle meet, and it is the center of the largest circle that fits inside the triangle touching all three sides.
Grade 10 – GeometryG-C.A.3Geometry and Measurement
Definition of Incenter
The incenter is the point inside a triangle that is the same distance from all three sides. Since any point on an angle bisector is equally far from the two sides of that angle, the bisectors from two corners cross at such a point, and the third bisector goes through it too.
That distance is the inradius \(r\). A circle centered at the incenter with radius \(r\) just touches each side, so it is the inscribed circle. Unlike the circumcenter, the incenter is always inside the triangle, whatever its shape.
Formula and notation
If \(I\) is the incenter and the circle touches the sides at \(X\), \(Y\), \(Z\), then \(IX = IY = IZ = r\). The inradius from the area \(K\) and semiperimeter \(s\) is:
\[ r = \frac{K}{s}, \qquad s = \frac{a + b + c}{2} \]
Here \(a\), \(b\), \(c\) are the side lengths.
Key facts
- It always lies inside the triangle, for acute, right and obtuse shapes alike.
- It is equally far from the three sides, not from the three vertices.
- Its circle touches every side exactly once.
- The inradius equals the triangle’s area divided by its semiperimeter.
Incenter example
A triangular tile has sides 7 in., 24 in. and 25 in. Find the radius of the largest circle that fits inside it.
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The sides satisfy \(7^2 + 24^2 = 625 = 25^2\), so it is a right triangle with legs 7 and 24. Area: \(K = \frac{1}{2}(7)(24) = 84\).
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Semiperimeter: \(s = \frac{7 + 24 + 25}{2} = 28\).
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Inradius: \(r = \frac{84}{28} = 3\).
The largest circle has radius 3 in., centered at the incenter.
Common mistake
Watch out
Bea uses the perimeter instead of the semiperimeter and computes \(r = \frac{84}{56} = 1.5\) in.
Why it is wrong
The formula divides the area by the semiperimeter, half the perimeter. Here \(s = 28\), not 56, so \(r = 3\) in. A check: for a right triangle, \(r = \frac{7 + 24 – 25}{2} = 3\) as well.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement