A scalene triangle has three sides of different lengths, and all angles of different measures.

Variables

a,b,c
→
Sides
α, β, γ
→
Angles
ha, hb, hc
→
Altitudes
ma, mb, mc
→
Medians
R
→
Radius of circumscribed circle
r
→
Radiu of inscribed circle
A
→
Area
p
→
Semiperimeter
1.
Sum of interior angles
$\alpha + \beta + \gamma = 180\degree$
Angle
Triangle
2.
Triangle inequality
$\begin{aligned} a+b&>c \\ a+c&>b\\b+c&>a \end{aligned}$
Side
Triangle
3.
Triangle inequality
$\begin{aligned}|a-b| <& c \\ |a-c| <& b \\ |b-c| <& a \end{aligned}$
Side
Triangle
4.
Midline of a triangle
$q=\dfrac{a}{2}, q||a$
Triangle
5.
Law of cosinse for side a
$a^2=b^2+c^2-2bc\cos\gamma$
Triangle
6.
Law of cosinse for side b
$b^2=a^2+c^2-2ac\cos\beta$
Triangle
7.
Law of cosinse for side c
$c^2 = a^2 + b^2 - 2ab\cos\gamma$
Triangle
8.
Law of sines
$\dfrac{a}{\sin\alpha}=\dfrac{b}{\sin\beta}=\dfrac{c}{\sin\gamma}=2R$
Triangle
9.
Circumscribed circle radius
$R=\dfrac{a}{2\sin\alpha}=\dfrac{b}{2\sin\beta}=\dfrac{c}{2\sin\gamma}$
Triangle
10.
Circumscribed circle radius
$R=\dfrac{bc}{2h_a} = \dfrac{ac}{2h_b} = \dfrac{bc}{2h_c}$
Triangle
11.
Inscribed circle radius
$r^2=\dfrac{(p-a)(p-b)(p-c)}{p}, \,\,\text{ and }\,\, p=\dfrac{a+b+c}{2}$
Triangle
12.
Inscribed circle radius
$r=\dfrac{1}{ha}+\dfrac{1}{h_b}+\dfrac{1}{h_c}$
Triangle
13.
Sine of α/2
$\sin\dfrac{\alpha}{2} = \sqrt{\dfrac{(p-b)(p-c)}{bc}}, \text{ where } p=\dfrac{a+b+c}{2}$
Triangle
Trigonometry
14.
Cosine α/2
$\cos\dfrac{\alpha}{2} = \sqrt{\dfrac{p(p-a)}{bc}}, \text{ where } p=\dfrac{a+b+c}{2}$
Triangle
Trigonometry
15.
Tangent α/2
$\tan \dfrac{\alpha}{2}=\sqrt{\dfrac{(p-b)(p-c)}{p(p-a)}}, \text{ where } p=\dfrac{a+b+c}{2}$
Triangle
Trigonometry
16.
Triangle height ha
$h_a=\dfrac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}$
Triangle
Trigonometry
17.
Triangle height ha
$h_a=b\sin\gamma$
Triangle
Trigonometry
18.
Median of a triangle
$m_a=\dfrac{b^2+c^2}{2}-\dfrac{a^2}{2}$
Triangle
Trigonometry
19.
Tirangle area - verison 1
$A=\dfrac{ah_a}{2}$
Area
Triangle
20.
Triangle area - version 2
$A = \dfrac{ab\sin\gamma}{2}$
Area
Triangle
Trigonometry
21.
Hereons Formula for triangle area
$A = \sqrt{p(p-a)(p-b)(p-c}, \text{ where } p = \dfrac{a+b+c}{2}$
Area
Triangle
22.
Triangle area - version 3
$A= pr, \text{ where } p = \dfrac{a+b+c}{2}$
Area
Triangle
23.
Triangle area - version 4
$A = \dfrac{abc}{4R}$
Angle
Triangle
24.
Triangle area - version 5
$A = 2R^2\sin\alpha \sin\beta \sin\gamma$
Area
Triangle
Trigonometry
25.
Tirangle area - version 6
$A = p^2 \tan\dfrac{\alpha}{2}\tan\dfrac{\beta}{2}\tan\dfrac{\gamma}{2}$
Area
Triangle

© 2024 Milos Petrovic – All Rights Reserved