The document discusses various properties and formulas related to triangles. It covers topics like the circumcenter, sine rule, cosine rule, projection rule, tangent rule, area formulas, relationships between angles and sides, and properties related to isosceles, right, and equilateral triangles. In total it presents over 20 different properties and formulas for working with triangles.

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Properties of Triangles ( by tarun gehlot)
1.
The perpendicular bisectors of the sides of a triangle are concurrent. The point of concurrence is
called circumcentre of the triangle. If S is the circumcentre of ΔABC, then SA = SB = SC. The
circle with center S and radius SA passes through the three vertices A, B, C of the triangle. This
circle is called circumcircle of the triangle. The radius of the circumcircle of ΔABC is called
circumradius and it is denoted by R.
2.
Sine Rule :
a
b
c
=
=
= 2R.
sin A sin B sin C
∴ a = 2R sin A, b = 2R sin B, c = 2R sin C.
3.
Cosine Rule : a2 = b2 + c2 – 2bc cos A, b2 = c2 + a2 – 2ca cos B, c2 = a2 + b2 – 2ab cos C.
4.
cos A =
b2 + c 2 − a 2
2bc
, cos B =
cos C =
a2 + b2 − c 2
2ab
.
c 2 + a2 − b2
2ca
,
5.
Projection Rule : a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A.
6.
Tangent Rule or Napier’s Analogy : tan⎛
⎜
A
B−C⎞ b−c
cot ,
⎟=
2
2 ⎠ b+c
⎝
B
⎛C−A ⎞ c −a
cot ,
tan⎜
⎟=
2
⎝ 2 ⎠ c+a
C
⎛ A −B⎞ a −b
tan⎜
cot .
⎟=
2
⎝ 2 ⎠ a+b
7.
Mollweide Rule :
a+b
=
c
⎛ A −B⎞
⎛ A −B⎞
cos⎜
sin⎜
⎟
⎟
2 ⎠ a−b
⎝
⎝ 2 ⎠
=
,
C
C
c
sin
cos
2
2
(s − b)(s − c )
B
, sin =
bc
2
8.
sin
A
=
2
9.
cos
A
=
2
s( s − a)
B
, cos =
bc
2
10. tan
A
=
2
( s − b)(s − c )
s(s − a)
, tan
(s − c )(s − a)
C
, sin =
ca
2
s(s − b)
C
, cos =
ca
2
B
=
2
(s − c )(s − a)
s( s − b)
(s − a)(s − b)
.
ab
s( s − c )
.
ab
, tan
1
C
=
2
(s − a)(s − b)
s(s − c )

2. Properties of Triangles
11. tan
A
Δ
( s − b)(s − c )
=
=
,
2 s(s − a)
Δ
tan
B
Δ
(s − c )(s − a)
,
=
=
Δ
2 s( s − b)
tan
C
Δ
(s − a)(s − b)
=
=
Δ
2 s(s − c )
12. cot
.
A s(s − a)
B s(s − b)
C s(s − c )
, cot =
, cot =
=
2
2
Δ
Δ
Δ
2
13. Area of ΔABC is Δ =
1
1
1
2
bc sin A = ca sin B = sin C = 2R sin A sin B sin C =
2
2
2
abc
s( s − a)(s − b)(s − c ) .
4R
14. r =
B
C
Δ
A
A
B
C
= (s − a) tan = (s − b) tan = (s − c ) tan =
= 4R sin sin sin
2
2
2
2
2
2
s
a
cot
B
+ cot
C
2
C
=
cot
A
+ cot
2
B
2
15. r1 =
Δ
A
B
C
A
B
C
= 4R sin cos cos = s tan
= (s − b) cot = (s − c ) cot =
s−a
2
2
2
2
2
2
16. r2 =
Δ
C
A
B
=
= s tan = (s − c ) cot = (s − a) cot
2
2
2
s−b
4R cos
17. r3 =
A
B
C
sin cos =
2
2
2
b
tan
A
C
+ tan
2
2
.
A
B
C
Δ
=
= s tan = (s − a) cot = (s − b) cot
2
2
2
s−c
c
.
B
A
tan + tan
2
2
18.
1 1 1 1
+ + = .
r1 r2 r3 r
19. r r1 r2 r3 = Δ2.
∑ a sin(B − C) = 0 .
ii) ∑ a cos(B − C) = 3abc
20. i)
3
3
iii) a2 sin 2B + b2 sin 2A = 4Δ
2
a
.
C
B
tan + tan
2
2
2
b
=
cot
C
2
+ cot
A
2

3. Properties of Triangles
a2 + b2 + c 2
4Δ
21. i) cotA + cotB + cotC =
ii) cot
A
B
C (a + b + c )2
.
cot cot =
2
2
2
4Δ
22. i) If a cos B = b cos A, then the triangle is isosceles.
ii) If a cos A = b cos B, then the triangle is isosceles or right angled.
iii)If a2 + b2 + c2 = 8R2, then the triangle is right angled.
iv) If cos2A + cos2B + cos2C = 1, then the triangle is right angled.
v) If cosA =
vi) If
sin B
, then the triangle is isosceles.
2 sin C
a
b
c
, then the triangle is equilateral.
=
=
cos A cos B cos C
vii) If cosA + cosB + cosC = 3/2, then the triangle is equilateral.
viii) If sinA + sinB + sinC =
3 3
, then the triangle is equilateral.
2
ix) If cotA + cotB + cotC = 3 , then the triangle is equilateral.
23. i) If
a2 + b2
a −b
2
2
=
sin( A + B)
, then C = 90°.
sin( A − B)
ii) If
a+b
b
= 1, then C = 60°.
+
b+c c+a
iii)If
1
1
3
, then A = 60°
+
=
a+b a+c a+b+c
iv) If
b
a2 − c 2
+
c
a2 − b 2
= 0, then A = 60°.
C
B
A
are in H.P.
, sin2 , sin2
2
2
2
i)
a, b, c are In H.P. ⇔ sin2
ii)
a, b, c are in A.P. ⇔ cot
B
A
C
, cot , cot
2
2
2
iii)
a, b, c are in A.P. ⇔ tan
A
B
C
are in H.P.
, tan , tan
2
2
2
iv)
a2, b2, c2 are in A.P. ⇔ cotA, cotB, cotC are in A.P.
v)
a2, b2, c2 are in A.P.
⇔
are in A.P.
tanA, tanB, tanC are in H.P
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