Propeties of-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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