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Incenter

Geometry is a very important branch that studies about the geometrical shapes and their properties. A triangle is a very basic geometrical figure which is seen everywhere in mathematics. It is a two-dimensional closed figure bounded by three line segments. There are various important concepts related to circle, such as - incenter, cicumcenter, orthocenter etc. In this article, we are going to study about incenter of a triangle.

Incenter of a triangle is defined as the point of intersection of the internal bisectors of a triangle. By internal bisectors, we mean the angle bisectors of interior angles of a triangle. Since there are three interior angles in a triangle, there must be three internal bisectors. The intersection point of all three internal bisectors is known as incenter of a circle. Also, if a perpendicular is drawn from the incenter to any of the side; and taking this distance as the radius and assuming incenter as center, a circle is drawn. This circle is known as incircle of the given triangle. In the triangle ABC shown below, when we draw bisectors for all the three angles, the three bisectors meet each other at a point. This point is called as the incenter of the triangle. The three lines are the bisectors of the triangle and they meet at a point r, which is the incenter of the triangle.

Incenter

 

How to Find the Incenter?

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Given below are the steps to find the incenter of a triangle:

1. Draw a triangle of any dimension

Create an Triangle

2. Using a compass, draw two arcs on two successive sides of the triangle from a vertex. There are no defined width for the arcs.

Bisect an Angle of Triangle

3. Without changing the compass reading, draw another two intersecting arcs inside the triangle from the arcs previously drawn. Now, we get an intersecting point inside the triangle.

Solve Incenter

4. Join the vertex and the point of intersection between the arcs. The line joining the vertex and the point of intersection is called as the angular bisector of the corresponding vertex.

Solve Incenter of Triangle

5. Again using the compass, draw two arcs of any length on the sides of the triangle from next vertex of the triangle.

Fine Incenter of Triangle

6. Without changing the compass reading, draw another two intersecting arcs inside the triangle from the arcs previously drawn. Now, we get an intersecting point inside the triangle.

Bisect Two Angles of Triangle

7. Join the corresponding vertex and the point of intersection between the arcs. The line is the angular bisector of the vertex.

How to Find Incenter

8. Now, the point of intersection between the two angular bisectors is called as the incenter of the triangle. Thus, the incenter of the triangle is solved using angular bisector.

Triangle Incenter

Properties of Incenter

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Listed below are the Properties of Incenter:

  • Incenter is equidistant from all the sides of the triangle.
  • The biggest circle that can be drawn inside a triangle is the incircle. The incircle will touch all sides of the triangle. The radius of the circle is the length of the perpendicular line drawn from the Incenter to any side.

Properties Of Incenter

  • For an equilateral triangle, incentre, circumcentre and centroid will all be the same. This is because the internal bisector is also a perpendicular bisector for the opposite side.

Incenter Pictures

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Given below are the pictures of incentres of acute, right angled and obtuse triangle. Unlike circumcentre, incenter always lies inside the triangle.

See the incentres 'D'of all the three types of triangles. Incenter Of acute Angled Triangle Incenter Of Right Angled Triangle

Incenter Of Obtuse Angled Triangle

Theorem on Incenter of Triangle

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Incenter is obtained by the Intersection of a triangle's three angle bisectors and it is equidistant from the three sides of the triangle.

To Prove: Incenter is equidistant from the three sides of the triangle.

Theorem on Incentre of Triangle

The angle bisectors of a triangle pass through the same point.

In $\triangle$ ABC

AI, BI, CI are angle bisectors and IF $\perp$ AB, ID $\perp$ BC and IE $\perp$ AC.

Points on the angle bisector is equidistant from the sides of the angle

IF = ID and ID = IE

=> IF = IE (Transitive property of congruence)

So, I is on the angle bisector of angle A (Converse of angle bisector theorem)

Hence, I (Incenter) is equidistant from the sides of $\triangle$ ABC.

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