This is the third post of geometry concept series in which we will learn about the triangles, their types, their properties and important theorems and results. So what is a triangle?

It is a three-sided closed plane figure which is formed by joining three non-collinear points. It is denoted by the symbol △.

In the given △ABC

1. Sum of any two sides of a triangle is always greater than the third side.

2. Difference of any two sides of a triangle is always less than the third side.

3. Greater angle has greater side opposite to it and the smaller angle has smaller side opposite to it.

4. Let p, q and r be the three sides of a △ ABC and r is the largest side

1.

The perpendicular drawn from the opposite vertex of a side in a triangle is called an altitude or height.

The point of intersection of these three altitudes is called orthocentre.

In this triangle AE, CD and BF are the Altitudes and O is the orthocentre.

∠BOC = 180° - ∠BAC

∠COA = 180° - ∠ABC

∠AOB = 180° - ∠ACB

The line segment joining the mid-points of a side to the vertex opposite to the side is called a median.

In this triangle AE, CD and BF are the medians. The point of intersection of these three medians is called centroid.

A median bisects the area of a triangle i.e, ar (△ABE) = ar (△AEC) = ar (½ △ABC)

A centroid divides each median into the ratio of 2:1

∠BAE = ∠CAE = ½ ∠BAC

The point of intersection of these angles bisectors is called incentre.

In this triangle AE, CD and BF are the angle bisectors and O is the incentre.

The point of intersection of these perpendicular bisectors is called circumcentre.

DO, EO, FO are the perpendicular bisectors in the given triangle and O is the circumcentre.

OA = OB = OC (circumradius)

It is a three-sided closed plane figure which is formed by joining three non-collinear points. It is denoted by the symbol △.

In the given △ABC

- A, B and C are the three vertices of this △ABC.
- ∠A, ∠B and ∠C are interior angles of this triangle.
- ∠FCB, ∠CBE, ∠EBD, ∠DBA, ∠BAI, ∠IAH, ∠HAC, ∠ACG and ∠GCF.
- AB, BC and CA are the three sides of this triangle.
- Sum of the interior angles of a triangle is 180° i.e, ∠A + ∠B + ∠C = 180°
- Sum of three ordered exterior angles is 360°. That is ∠FCB + ∠DBA + ∠HAC = 360°. And similarly ∠ACG + ∠IAB + ∠CBE = 360°
- Sum of all the sides of this triangle is equal to its perimeter i.e, AB + BC + CA = Perimeter.
- Semiperimeter of a triangle is half of its perimeter.

**Types of Triangles**

All the triangles can be classified under two heads

- Interior angles
- Length of sides

__Triangles according to interior angles__**Acute angle triangle:**Each of the angles of a triangle is acute angle (i.e. less than 90°)**Right angled triangle:**One of the angles of a triangle is 90° and the remaining two angles are complementary to each other that means each of the remaining angles will be lees than 90°.**Obtuse angle triangle:**One of the angle is obtuse angle (i.e. more than 90°).

__Triangles according to length of sides__**Scalene Triangle:**All the sides and angles of this triangle are of different length.**Isosceles Triangle:**Two of the three sides of a triangle are equal. And also angles opposite equal angles are equal.**Equilateral Triangle:**All the sides and angles of this triangle are equal.

**Some Other Properties of Triangles**

1. Sum of any two sides of a triangle is always greater than the third side.

2. Difference of any two sides of a triangle is always less than the third side.

3. Greater angle has greater side opposite to it and the smaller angle has smaller side opposite to it.

4. Let p, q and r be the three sides of a △ ABC and r is the largest side

**.**Then

- If r² < p² + q², the triangle is acute angle triangle
- If r² = p² + q², the triangle is right angled triangle
- If r² > p² + q², the triangle is obtuse angle triangle

**Important Terms Associated With Triangles**

1.

**Altitude / Height and Orthocentre**

The perpendicular drawn from the opposite vertex of a side in a triangle is called an altitude or height.

The point of intersection of these three altitudes is called orthocentre.

In this triangle AE, CD and BF are the Altitudes and O is the orthocentre.

∠BOC = 180° - ∠BAC

∠COA = 180° - ∠ABC

∠AOB = 180° - ∠ACB

**2. Median and Centroid**

The line segment joining the mid-points of a side to the vertex opposite to the side is called a median.

In this triangle AE, CD and BF are the medians. The point of intersection of these three medians is called centroid.

A median bisects the area of a triangle i.e, ar (△ABE) = ar (△AEC) = ar (½ △ABC)

A centroid divides each median into the ratio of 2:1

**3. Angle Bisector and Incentre**

**A line segment which originates from a vertex and bisects the same angle is called an angle bisector.**

∠BAE = ∠CAE = ½ ∠BAC

The point of intersection of these angles bisectors is called incentre.

In this triangle AE, CD and BF are the angle bisectors and O is the incentre.

**4. Perpendicular Bisector and Cirumcentre**

**A line segment which bisects a side perpendicularly is called at a perpendicular bisector of a side of triangle.**

The point of intersection of these perpendicular bisectors is called circumcentre.

DO, EO, FO are the perpendicular bisectors in the given triangle and O is the circumcentre.

OA = OB = OC (circumradius)

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