•
•
•
•
•
•
•
•

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

The sides of a triangle are proportional to the sines of opposite angles i.e, In an ∆ABC, Where k is some constant.

LAWS OF COSINES OR COSINES RULE–

In any ∆ABC , We have PROJECTION FORMULA –

In any ∆ABC LAWS OF TANGENT OR TANGENT RULES (Napier’s Analogy) HALF ANGLE FORMULAE OR SEMI-SUM FORMULAE   AREA OF TRIANGLE –

The area of any triangle ABC can be given by: HERON’S FORMULA –

In a ∆ABC , if a+b+c = 2s, then it’s area can given by: CIRCUMCIRCLE OF A TRIANGLE In any triangle ABC,

a). b). Everywhere  means area

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 2). Area of a pedal triangle is 3). Length of the median AD, BE, and CF of the ∆ABC. is given by-  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   POINTS TO BE NOTED

1). Area of the cyclic quadrilateral ABCD is =  3). CosB = and similarly other angles.

PTOLEMY THEOREM –

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= 