Inside Story

- 1 PROPERTIES AND SOLUTIONS OF TRIANGLE
- 1.1 POINTS TO BE REMEMBERED
- 1.1.1 LAWS OF SINE OR SINE RULE
- 1.1.2 LAWS OF COSINES OR COSINES RULE–
- 1.1.3 PROJECTION FORMULA –
- 1.1.4 LAWS OF TANGENT OR TANGENT RULES (Napier’s Analogy)
- 1.1.5 HALF ANGLE FORMULAE OR SEMI-SUM FORMULAE
- 1.1.6 AREA OF TRIANGLE –
- 1.1.7 HERON’S FORMULA –
- 1.1.8 CIRCUMCIRCLE OF A TRIANGLE
- 1.1.9 INCIRCLE OF A TRIANGLE
- 1.1.10 EXCRIBED CIRCLE OF A TRIANGLE
- 1.1.11 ORTHOCENTRE OF A TRIANGLE
- 1.1.12 CYCLIC QUADRILATERAL
- 1.1.13 PTOLEMY THEOREM –
- 1.1.14 Was this helpful

- 1.1 POINTS TO BE REMEMBERED

**PROPERTIES AND SOLUTIONS OF TRIANGLE**

Here, there are some key concepts regarding the properties and solutions of triangles. these are very important for the students who preparing for IIT-JEE and other equivalent exams.

**POINTS TO BE REMEMBERED**

** LAWS OF SINE OR SINE RULE **

Where k is some constant.

**LAWS OF COSINES OR COSINES RULE**–

**PROJECTION FORMULA –**

In any ∆ABC

**LAWS OF TANGENT OR TANGENT RULES** **(Napier’s Analogy)**

**(Napier’s Analogy)**

**HALF ANGLE FORMULAE OR SEMI-SUM FORMULAE**

**AREA OF TRIANGLE** –

**HERON’S FORMULA – **

**CIRCUMCIRCLE OF A TRIANGLE**

In any triangle ABC,

**INCIRCLE OF A TRIANGLE**

In any triangle ABC,

a).

b).

c).

d).

**EXCRIBED CIRCLE OF A TRIANGLE**

In any triangle ABC

a).

b).

c).

d).

**ORTHOCENTRE OF A TRIANGLE**

The triangle formed by joining the points F, E, and D is called the **pedal triangle. **In any triangle ABC,

a). The distance of the orthocentre from the vertices of the triangle ABC is-

Distance from

Distance from

Distance from

b). Distance of the orthocentre from the sides of the triangle ABC is-

, ,

**POINTS TO BE NOTED**

1). The circumradius of a padel triangle is

2). Area of a pedal triangle is

3). **Length of the median AD, BE, and CF of the ∆ABC. is given by-**

AD =

BE =

CF =

4). Length of the angle bisector through vertex A=

Through vertex B =

Through vertex C =

5). Length of the altitude from;

From vertex A =

From vertex B =

From vertex C =

6). Distance between circumcentre and the orthocentre =

7). Distance between circumcentre and incentre is=

8). Distance between incentre and the orthocentre =

9). Distance between circumcentre and centroid =

10). If circumcentre, centroid, and orthocentre are collinear and G divides OO’ in the ratio of 1:2

11). If I1, I2, and I3 are the center of the escribed circles opposite to the angle A, B, and C respectively. And O is the orthocentre. Then

**CYCLIC QUADRILATERAL**

**POINTS TO BE NOTED**

1). Area of the cyclic quadrilateral ABCD is =

2). Circumradius of the cyclic quadrilateral =

3). CosB = and similarly other angles.

**PTOLEMY THEOREM – **

If ABCD is a cyclic quadrilateral then **AC.BD = AB.CD+BC.AD**

**REGULAR POLYGON**

**POINTS TO BE NOTED**

1). If a regular polygon has n sides, then the sum of its internal angle is (n-2)π and each angle is

2). In a regular polygon the centroid, the circumcentre, and incentre are the same.

3). Area of the regular polygon =

Where n is the no of sides of the regular polygon, R is the radius of circumscribing circle, and r is the radius of the incircle of a polygon.

4). Radius of circumscribing circle =

Where a is the length of the side of the regular polygon of n sides.

5). Radius of inscribed circles r=