# Centres of Triangle in Geometry

**There are four centres in a triangle:**

- In-centre
- Circum-centre
- Centroid
- Ortho centre

### In-centre:

The point of intersection of the all the three angle bisectors of a triangle is called as In-centre

The distance of the in-centre from the all the three sides is equal**(ID=IE=IF=inradius “r”)**

In-radius (r)= Area of triangle/Semiperimetre=A/S

**∠BIC = 90 +∠A/2**

**∠AIC=90+∠B/2**

**∠AIB=90+∠C/2**

### Circumcentre:

The point of intersection of the perpendicular bisectors of the three sides of a triangle is called its circumcentre.

The distance between the circumcentre and the three vertices of a triangle is always equal.

**OA=OB =OC=R(circumradius)=abc/4A**

**∠BOC=2∠A**

**∠AOC=2∠B**

**∠AOB=2∠C**

### Location of circumcentre in various types of triangle:

**Acute angle triangle: Lies inside the triangle**

**Obtuse angle triangle: Lies outside the triangle**

**Right angle triangle: Lies at the midpoint of the hypotenuse.**

### Orthocentre:

It is the point of intersection of all the three altitudes of the triangle.

**∠BHC=180-∠A**

**∠AHB=180-∠C**

**∠AHC=180-∠B**

### Position of orthocentre inside the triangle:

**Acute angled triangle: lies inside the triangle.**

**Obtuse angle triangle: lies outside the triangle on the backside of the obtuse angle. Orthocentre and circumcentre lie opposite to each other in obtuse angle triangle.**

**Right angle triangle: lies on the right angle of the triangle.**

### Centriod:

- It is the point of the intersection of the
**three median**of the triangle. It is denoted by G. - A centroid divides the area of the triangle in exactly
**three parts**.

### Medians:

- A line segment joining the
**midpoint**of the side with the opposite vertex is called median. - Median bisects the opposite side as well as divide the area of the triangle in
**two equal**parts.

### Some important tricks are as follows:

**(1)**In a right angle triangle ABC,∠B=90° & AC is the hypotenuse of the triangle. The perpendicular BD is dropped on the hypotenuse AC from the right angle vertex B,

**BD=AB*BC/AC**

**AD=AB ^{2}/AC**

**CD=BC ^{2}/AC**

**1/BD=1/AB ^{2} +1/BC^{2}**

**(2)**The ratio of the areas of the triangles with equal bases is equal to the ratio of their heights.

**(3)**The ratio of the areas of two triangles is equal to the ratio of products of base and its corresponding sides.

Area (Triangle ABC)/Area (Triangle PQR) =AC*BD/PR*QS

**(4)**If two triangles have the same base and lie between the same parallel lines then the area of the two triangles will be equal.

Area of Triangle ABC= Area of Triangle ADB

**(5)**In a triangle ABC, AE, CD and BF are the medians then

**3(AB ^{2}+BC^{2}+AC^{2}) = 4(CD^{2}+BF^{2}+AE^{2})**

**6)**Sum of any two sides of the triangle is always greater than the third sides.

**(7)**The difference of any two sides of a triangle is always less than the third sides.