How To Find the Center of the Circle Inscribed in a Triangle

Learn to find the center of the circle inscribed in a triangle. The bisectors of the angles

The bisectors of the angles of a triangle. The center of the inscribed circle



  • What is a bisector of an angle?

  • The bisector of an angle is a line segment inside the angle, with the origin at its apex, which forms with the sides of the angle two smaller congruent angles.
  • Important: Any point belonging to the bisector is equidistant from both sides of the angle.
  • Bisector of an angle

    Bisector of an angle

  • How to find the center of the circle inscribed in a triangle?

  • In any triangle the bisectors of the angles cross at the exact same point, which is the center of the circle tangent to the sides of the triangle, called the circle inscribed in the triangle. This is the largest circle that fits inside the triangle.
  • The bisectors and the circle inscribed in the triangle

    The bisectors and the circle inscribed in the triangle


  • In the triangle △ABC:
  • AA' is the bisector of the angle ∠BAC => ∠BAA' = ∠CAA' = ∠BAC / 2
  • BB' is the bisector of the angle ∠ABC => ∠ABB' = ∠CBB' = ∠ABC / 2
  • CC' is the bisector of the angle ∠ACB => ∠ACC' = ∠BCC' = ∠ACB / 2

Bisectors AA', BB' and CC' intersect at point O, which is the center of the circle inscribed in the triangle.


  • Why is the point of intersection of the bisectors the center of the circle inscribed in the triangle?

  • O is a point on the bisector AA' so it is equidistant from the line segments AB and AC =>
  • [1] the distance from the point O to the segment AB is OP, where OP ⊥ AB (⊥ is the symbol for perpendicular segments);
  • [2] the distance from the point O to the segment AC is ON, where ON ⊥ AC;
  • [3] OP ≅ ON.
  • O is a point on the bisector BB' so it is equidistant from the line segments AB and BC =>
  • [1] the distance from the point O to the segment AB is OP, where OP ⊥ AB;
  • [2] the distance from the point O to the segment BC is OM, where OM ⊥ BC;
  • [3] OP ≅ OM.
  • From the relations above, it follows that OP ⊥ AB, OM ⊥ BC, ON ⊥ AC and OM = ON = OP = r, where r is the radius of the circle inscribed in the triangle.

  • Notes:
  • A circle inscribed in a triangle is the largest circle that can be drawn inside a triangle so that it is tangent to all three of its sides.
  • A circle is tangent to a line when the line and the circle have exactly one point in common. The segment formed by the center of the circle and the point of contact with the line is perpendicular to the line.

» How To Calculate the Radius of the Circle Inscribed in a Triangle