Lesson 3 — Incircles
Big idea: Every triangle has a circle tangent to all three sides, centered where the angle bisectors meet.
Words to know
| Word | Meaning |
|---|---|
| Circle inscribed in a polygon | A circle is inscribed in a polygon iff each side of the polygon is tangent to the circle. |
| Polygon circumscribed about a circle | A polygon is circumscribed about a circle iff each of its sides is tangent to the circle. |
| Incircle | The circle inscribed in a polygon. |
| Incenter | The center of the incircle; where the angle bisectors meet. |
| Angle bisector | A ray that divides an angle into two equal angles. |
| Tangent | A line that touches a circle at exactly one point. |
| Equidistant | At the same distance from a point or line. |
Concepts to know
- Theorem 70: Every triangle has an incircle.
- Proof idea: the bisectors of two angles meet at P. Perpendiculars from P to the three sides are equal, so a circle centered at P is tangent to all three sides.
- Corollary to Theorem 70: The angle bisectors of a triangle are concurrent.
- The incenter is equidistant from the three sides.
- The inradius is the perpendicular distance from the incenter to any side.
- Construction 10: To inscribe a circle in a triangle, bisect two angles; the intersection is the center. Drop a perpendicular to a side to get the radius.
- Tangent segments drawn from the same outside point are equal.
- From the exercises: a quadrilateral has an incircle only if the sums of its opposite sides are equal.
Formulas / rules
- No new formula. The radius is the perpendicular distance from the incenter to a side.
- Incenter condition: distance to each side is the same.
- Incircle condition for a quadrilateral: a + c = b + d (sums of opposite sides).
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find the incenter | Bisect two angles of the triangle. |
| Find the inradius | Drop a perpendicular from the incenter to any side and measure it. |
| Check a point | Verify equal perpendicular distances to all three sides. |
| Construct the incircle | Find the incenter, then draw a perpendicular to one side for the radius. |
| Compare incircle and circumcircle | Incircle uses angle bisectors; circumcircle uses perpendicular bisectors. |
| Work with tangent segments | Mark equal tangent lengths from each vertex. |
Watch out
- The incenter is always inside the triangle.
- Do not use perpendicular bisectors to find the incenter.
- A tangent is perpendicular to the radius at the point of tangency.
- “Inscribed” and “circumscribed” depend on which figure you name first.
- The incenter is equidistant from the sides, not the vertices.