## 28 Apr Properties of triangles 5

At Miracle Learning Centre, we teach you interesting stuff in math tuition. In our previous maths tuitions, we have already explored some important properties of triangles. In this tutorial, we are going to explore some more interesting properties of triangles. Today, in this maths tuition, we are going to learn about the orthocenter, the in-centre and the circum centre of a triangle. If you have not gone through our previous blogs related to triangles, it is highly recommended that you read those articles and develop a solid understanding of the concepts discussed there before going through this article.

### The Orthocenter

For any triangle, the orthocenter is the point where three altitudes of a triangle meet. Let us see this with an example. Consider the following triangle ABC:

In the above triangle ABC, AD is the height or altitude of the triangle. By definition of height, since AD is the height or altitude of the triangle, AD must be perpendicular to the side BC.

BE and CF are also heights of the triangle ABC. AD, BE and CF all meet at the point O which is known as the Orthocenter of a triangle.

Remember that when O is the Orthocenter of a triangle, then

Angle BOC + Angle BAC = 1800

And similarly:

Angle AOC + Angle ABC = 1800

And Angle AOB + Angle ACB = 1800

**The Incenter**

For any triangle, the incentre is the point where the three angle bisectors meet. Let us see this with an example. Consider the following triangle PQR:

P

U

T

S

Q

R

I

In the above triangle PQR, the line segment PS divides the angle QPR into two equal angles. That is:

Angle SPQ = Angle SPR

The line segment PS is known as the angle bisector of the angle QPR.

Similarly, the line segment RU is the angle bisector of the angle QRP and QT is the angle bisector of the angle PQR.

I is the point where all these points meet and I is known as the incentre. If we drew a circle inside the triangle PQR is such a way that the circle touched each side of the triangle at exactly one point, then incenter I would have been the center of such a circle.

Remember that:

Angle QIR = 900 + ½ of (Angle QPR)

And similarly:

Angle PIR = 900 + ½ of (Angle PQR) and

Angle PIQ = 900 + ½ of (Angle PRQ)

The Circumcenter

For any triangle, the circumcenter is the point where three perpendicular side bisectors meet. Let us see this with an example.

O

R

Q

P

L

K

G

F

E

D

C

B

A

In the triangle ABC above, the line segment DE bisects BC i.e. divides it equally into two parts BP and PC. As a result, P is the mid-point of BC. DE is also perpendicular to BC. Thus, DE, in this case is the perpendicular bisector of the side BC.

Similarly, FG is the perpendicular bisector of AB and KL is the perpendicular bisector of AC. The three perpendicular bisectors meet at the point O which is known as the circumcenter. If we drew a circle around the triangle ABC in such a way that the circle touched all the three vertices of the triangle, then O would have been the center of such a circle.

Remember that:

Angle BOC = 2. (Angle BAC)

And similarly:

Angle AOC = 2. (Angle ABC) and

Angle AOB = 2. (Angle ACB)

**For equilateral triangles, the incenter, the circumcentre, the centroid and the orhocentre is the same point.** It is a homework for you to find out why that is the case. In case you cannot figure out yourself, you can seek help from your peers or teachers at your math tuition.

So in this math tuition, we have taken a look at some more properties of triangles. A thorough understanding of the concepts discussed here is necessary to build a solid foundation in geometry and hopefully, this article will help you to achieve that. We hope you enjoyed this math tuition from Miracle Learning Centre. Look out for the next math tuition from Miracle Learning Centre.

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