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
α+β+γ=180°\alpha + \beta + \gamma = 180\degree
Angle
Triangle
2.
Triangle inequality
a+b>ca+c>bb+c>a\begin{aligned} a+b&>c \\ a+c&>b\\b+c&>a \end{aligned}
Side
Triangle
3.
Triangle inequality
∣a−b∣<c∣a−c∣<b∣b−c∣<a\begin{aligned}|a-b| <& c \\ |a-c| <& b \\ |b-c| <& a \end{aligned}
Side
Triangle
4.
Midline of a triangle
q=a2,q∣∣aq=\dfrac{a}{2}, q||a
Triangle
5.
Law of cosinse for side a
a2=b2+c2−2bccos⁡γa^2=b^2+c^2-2bc\cos\gamma
Triangle
6.
Law of cosinse for side b
b2=a2+c2−2accos⁡βb^2=a^2+c^2-2ac\cos\beta
Triangle
7.
Law of cosinse for side c
c2=a2+b2−2abcos⁡γc^2 = a^2 + b^2 - 2ab\cos\gamma
Triangle
8.
Law of sines
asin⁡α=bsin⁡β=csin⁡γ=2R\dfrac{a}{\sin\alpha}=\dfrac{b}{\sin\beta}=\dfrac{c}{\sin\gamma}=2R
Triangle
9.
Circumscribed circle radius
R=a2sin⁡α=b2sin⁡β=c2sin⁡γR=\dfrac{a}{2\sin\alpha}=\dfrac{b}{2\sin\beta}=\dfrac{c}{2\sin\gamma}
Triangle
10.
Circumscribed circle radius
R=bc2ha=ac2hb=bc2hcR=\dfrac{bc}{2h_a} = \dfrac{ac}{2h_b} = \dfrac{bc}{2h_c}
Triangle
11.
Inscribed circle radius
r2=(p−a)(p−b)(p−c)p,   and   p=a+b+c2r^2=\dfrac{(p-a)(p-b)(p-c)}{p}, \,\,\text{ and }\,\, p=\dfrac{a+b+c}{2}
Triangle
12.
Inscribed circle radius
r=1ha+1hb+1hcr=\dfrac{1}{ha}+\dfrac{1}{h_b}+\dfrac{1}{h_c}
Triangle
13.
Sine of α/2
sin⁡α2=(p−b)(p−c)bc, where p=a+b+c2\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⁡α2=p(p−a)bc, where p=a+b+c2\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⁡α2=(p−b)(p−c)p(p−a), where p=a+b+c2\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
ha=2ap(p−a)(p−b)(p−c)h_a=\dfrac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}
Triangle
Trigonometry
17.
Triangle height ha
ha=bsin⁡γh_a=b\sin\gamma
Triangle
Trigonometry
18.
Median of a triangle
ma=b2+c22−a22m_a=\dfrac{b^2+c^2}{2}-\dfrac{a^2}{2}
Triangle
Trigonometry
19.
Tirangle area - verison 1
A=aha2A=\dfrac{ah_a}{2}
Area
Triangle
20.
Triangle area - version 2
A=absin⁡γ2A = \dfrac{ab\sin\gamma}{2}
Area
Triangle
Trigonometry
21.
Hereons Formula for triangle area
A=p(p−a)(p−b)(p−c, where p=a+b+c2A = \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, where p=a+b+c2A= pr, \text{ where } p = \dfrac{a+b+c}{2}
Area
Triangle
23.
Triangle area - version 4
A=abc4RA = \dfrac{abc}{4R}
Angle
Triangle
24.
Triangle area - version 5
A=2R2sin⁡αsin⁡βsin⁡γA = 2R^2\sin\alpha \sin\beta \sin\gamma
Area
Triangle
Trigonometry
25.
Tirangle area - version 6
A=p2tan⁡α2tan⁡β2tan⁡γ2A = p^2 \tan\dfrac{\alpha}{2}\tan\dfrac{\beta}{2}\tan\dfrac{\gamma}{2}
Area
Triangle

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