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Search results:
421.
Triangle inequality
a
+
b
>
c
a
+
c
>
b
b
+
c
>
a
\begin{aligned} a+b&>c \\ a+c&>b\\b+c&>a \end{aligned}
a
+
b
a
+
c
b
+
c
>
c
>
b
>
a
Side
Triangle
422.
Triangle inequality
∣
a
−
b
∣
<
c
∣
a
−
c
∣
<
b
∣
b
−
c
∣
<
a
\begin{aligned}|a-b| <& c \\ |a-c| <& b \\ |b-c| <& a \end{aligned}
∣
a
−
b
∣
<
∣
a
−
c
∣
<
∣
b
−
c
∣
<
c
b
a
Side
Triangle
423.
Midline of a triangle
q
=
a
2
,
q
∣
∣
a
q=\dfrac{a}{2}, q||a
q
=
2
a
,
q
∣∣
a
Triangle
424.
Law of cosinse for side a
a
2
=
b
2
+
c
2
−
2
b
c
cos
γ
a^2=b^2+c^2-2bc\cos\gamma
a
2
=
b
2
+
c
2
−
2
b
c
cos
γ
Triangle
425.
Law of cosinse for side b
b
2
=
a
2
+
c
2
−
2
a
c
cos
β
b^2=a^2+c^2-2ac\cos\beta
b
2
=
a
2
+
c
2
−
2
a
c
cos
β
Triangle
426.
Law of cosinse for side c
c
2
=
a
2
+
b
2
−
2
a
b
cos
γ
c^2 = a^2 + b^2 - 2ab\cos\gamma
c
2
=
a
2
+
b
2
−
2
ab
cos
γ
Triangle
427.
Law of sines
a
sin
α
=
b
sin
β
=
c
sin
γ
=
2
R
\dfrac{a}{\sin\alpha}=\dfrac{b}{\sin\beta}=\dfrac{c}{\sin\gamma}=2R
sin
α
a
=
sin
β
b
=
sin
γ
c
=
2
R
Triangle
Variables:
R
→
Radius of circumscribed circle
428.
Circumscribed circle radius
R
=
a
2
sin
α
=
b
2
sin
β
=
c
2
sin
γ
R=\dfrac{a}{2\sin\alpha}=\dfrac{b}{2\sin\beta}=\dfrac{c}{2\sin\gamma}
R
=
2
sin
α
a
=
2
sin
β
b
=
2
sin
γ
c
Triangle
429.
Circumscribed circle radius
R
=
b
c
2
h
a
=
a
c
2
h
b
=
b
c
2
h
c
R=\dfrac{bc}{2h_a} = \dfrac{ac}{2h_b} = \dfrac{bc}{2h_c}
R
=
2
h
a
b
c
=
2
h
b
a
c
=
2
h
c
b
c
Triangle
Variables:
ha, hb, hc
→
Altitudes
a, b, c
→
Sides
430.
Inscribed circle radius
r
2
=
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
p
,
and
p
=
a
+
b
+
c
2
r^2=\dfrac{(p-a)(p-b)(p-c)}{p}, \,\,\text{ and }\,\, p=\dfrac{a+b+c}{2}
r
2
=
p
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
,
and
p
=
2
a
+
b
+
c
Triangle
Variables:
p
→
Semiperimeter
431.
Inscribed circle radius
r
=
1
h
a
+
1
h
b
+
1
h
c
r=\dfrac{1}{ha}+\dfrac{1}{h_b}+\dfrac{1}{h_c}
r
=
ha
1
+
h
b
1
+
h
c
1
Triangle
Variables:
ha, hb, hc
→
Altitudes
432.
Sine of α/2
sin
α
2
=
(
p
−
b
)
(
p
−
c
)
b
c
,
where
p
=
a
+
b
+
c
2
\sin\dfrac{\alpha}{2} = \sqrt{\dfrac{(p-b)(p-c)}{bc}}, \text{ where } p=\dfrac{a+b+c}{2}
sin
2
α
=
b
c
(
p
−
b
)
(
p
−
c
)
,
where
p
=
2
a
+
b
+
c
Triangle
Trigonometry
Variables:
p
→
Semiperimeter
a,b,c
→
Sides
α
→
Angle
433.
Cosine α/2
cos
α
2
=
p
(
p
−
a
)
b
c
,
where
p
=
a
+
b
+
c
2
\cos\dfrac{\alpha}{2} = \sqrt{\dfrac{p(p-a)}{bc}}, \text{ where } p=\dfrac{a+b+c}{2}
cos
2
α
=
b
c
p
(
p
−
a
)
,
where
p
=
2
a
+
b
+
c
Triangle
Trigonometry
Variables:
p
→
Semiperimeter
434.
Tangent α/2
tan
α
2
=
(
p
−
b
)
(
p
−
c
)
p
(
p
−
a
)
,
where
p
=
a
+
b
+
c
2
\tan \dfrac{\alpha}{2}=\sqrt{\dfrac{(p-b)(p-c)}{p(p-a)}}, \text{ where } p=\dfrac{a+b+c}{2}
tan
2
α
=
p
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
,
where
p
=
2
a
+
b
+
c
Triangle
Trigonometry
Variables:
p
→
Semiperimeter
435.
Triangle height ha
h
a
=
2
a
p
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
h_a=\dfrac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}
h
a
=
a
2
p
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
Triangle
Trigonometry
Variables:
p
→
Semiperimeter
436.
Triangle height ha
h
a
=
b
sin
γ
h_a=b\sin\gamma
h
a
=
b
sin
γ
Triangle
Trigonometry
437.
Median of a triangle
m
a
=
b
2
+
c
2
2
−
a
2
2
m_a=\dfrac{b^2+c^2}{2}-\dfrac{a^2}{2}
m
a
=
2
b
2
+
c
2
−
2
a
2
Triangle
Trigonometry
Variables:
ma
→
Medina to side a
438.
Tirangle area - verison 1
A
=
a
h
a
2
A=\dfrac{ah_a}{2}
A
=
2
a
h
a
Area
Triangle
Variables:
A
→
Area
a
→
Side
ha
→
Height to side a
439.
Triangle area - version 2
A
=
a
b
sin
γ
2
A = \dfrac{ab\sin\gamma}{2}
A
=
2
ab
sin
γ
Area
Triangle
Trigonometry
Variables:
A
→
Area
a,b
→
Sides
γ
→
Angle between sides a and b
440.
Hereons Formula for triangle area
A
=
p
(
p
−
a
)
(
p
−
b
)
(
p
−
c
,
where
p
=
a
+
b
+
c
2
A = \sqrt{p(p-a)(p-b)(p-c}, \text{ where } p = \dfrac{a+b+c}{2}
A
=
p
(
p
−
a
)
(
p
−
b
)
(
p
−
c
,
where
p
=
2
a
+
b
+
c
Area
Triangle
Variables:
A
→
Area
p
→
Semiperimeter
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Isoscales triangle
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Most Important Formulas
Sum and difference formulas
Double angle formulas
Half-angle formulas
Multiple Angle Formulas
Sum to product
Product To Sum
Power of trigonometric functions
Two - dimensional coordinate system
Line in plane
Circle
Ellipse
Hyperbola
Limits of Functions
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Differentiation Formulas
Derivatives of Composite Functions
Higher Order Derivatives
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