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- In the triangle ABC, if â€˜Râ€™ is the circum radius then, R = a/2 sin A = b/2 sin B = c/2 sin C = abc/4Î”.
- In case of an in-circle of triangle ABC, if â€˜râ€™ is the radius of the in-circle, then r = Î”/s

= (s â€“ a) tan A/2

= (s â€“ b) tan B/2

= (s â€“ c) tan C/2

= [a sin (B/2) sin (C/2)]/ cos A/2

= [b sin (A/2) sin (C/2)]/ cos B/2

= [c sin (B/2) sin (A/2)]/ cos C/2

= 4R sin A/2 sin B/2 sin C/2 - The relation between the radius of in circle and circum circle is given by the inequality 2r â‰¤ R.
- The above inequality reduces into equality only in case of an equilateral triangle.
- If r
_{1}, r_{2}and r_{3}are the radii of the escribed circles opposite to the angles A, B and C then,

1. r_{1 }= Î”/s-a, r_{2}= Î”/s-b, r_{3}= Î”/s-c

2. r_{1}= s tan A/2, r_{2}= s tan B/2, r_{3}= s tan C/2

3. r_{1}= [a cos (B/2) cos (C/2)]/ cos A/2

4. r_{2 }= [b cos (C/2) cos (A/2)]/ cos B/2

5. r_{3}= [c cos (A/2) cos (B/2)]/ cos C/2 - Circum-center of the pedal triangle of a given triangle bisects the line joining the circum-center of the triangle to the orthocenter.
- Orthocenter of a triangle is the same as the in-centre of the pedal triangle in the same triangle.
- If I
_{1}, I_{2}and I_{3}are the centers of the escribed circles which are opposite to A, B and C respectively and I is the center of the in-circle, then triangle ABC is the pedal triangle of the triangle I_{1}I_{2}I_{3}and I is the orthocenter of the triangle I_{1}I_{2}I_{3}. - The centroid of the triangle lies on the line joining the circum center to the orthocenter and divides it in the ratio 1: 2.
- If â€˜Oâ€™ is the orthocenter and DEF is the pedal triangle of Î”ABC, where AD, BE and CF are the perpendiculars drawn from A, B and C to the opposite sides, then

1. OA = 2R cos A

2. OB = 2R cos B

3. OC = 2R cos C

4. The circum radius of the pedal triangle = R/2

5. The area of the pedal triangle is = 2Î” cos A cos B cos C **Some Important Results:**

1. tan A/2 tan B/2 = (s-c)/s

2. tan A/2 + tan B/2 = c/s cot C/2 = c(s-c)/Î”

3. tan A/2 - tan B/2 = (a-b)(s-c)/Î”

4. cot A/2 + cot B/2 = (tan A/2 + tan B/2)/ (tan A/2.tan B/2)

= c/(s-c) cot C/2

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