Triangle - basic concepts

$A,B,C$ - vertices
$a=BC, b=AC, c=AB$ - sides
$α, β, γ$ - interior angles
$α', β', γ'$ - exterior angles

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The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

$$a+b>c$$ $$b+c>a$$ $$a+c>b$$

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The sum of interior angles is 180°

$$α+β+γ=180°$$

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The sum of the exterior angles is 360°

$$α'+β'+γ'=360°$$

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The exterior angle is equal to the sum of opposite interior angles

$$α'=β+γ$$ $$β'=α+γ$$ $$γ'=α+β$$

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The perimeter of the triangle is equal to the sum of the three sides

$$P=a+b+c$$

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Heights

$h_a, h_b, h_c$ - heights;
$O$ - orthocenter;

The sides relate to each other as to their heights.

$$a:b:c = 1/h_a:1/h_b:1/h_c$$

Medians

$P, M, N$ - middles of the sides
$m_a, m_b, m_c$ - medians
$O$ - centroid

The centroid divides the medians in the ratio 2:1

$$NO:OB = 1:2$$ $$MO:OA = 1:2$$ $$PO:OC = 1:2$$

Finding the median by the sides of the triangle

$$m_a = 1/2√{2(b^2+c^2)-a^2}$$

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Calculate side with the medians

$$a=2/3√{2(m_c^2+m_b^2)-m_a^2}$$ $$b=2/3√{2(m_c^2+m_a^2)-m_b^2}$$ $$c=2/3√{2(m_a^2+m_b^2)-m_c^2}$$

Bisectors

$l_a, l_b, l_c$ - bisectors of internal angles
$O$ - incenter

$$l_a=2/{b+c}√{bcp(p-a)}$$

The ratio in which the bisector divides the side is equal to the ratio of the other two sides. The same goes for the bisectors of external angles.

$$AM : MB = AC : BC$$

Bisectors

$S_a, S_b, S_c$ - bisectors
$O$ - circumcenter

$$AO=BO=CO=R$$ $$R = {bc}/2h_a = {ac}/2h_b = {ab}/2h_c$$

Central median

$M$ - midpoint of $AC$
$N$ - midpoint of $BC$
$MN$ - central median

$$MN ∥ AB$$ $$MN = 1/2AB$$

Types of triangles

Isosceles triangle

$$a=b$$ $$α=β$$ $$h_c≡m_c≡l_c$$

Equilateral triangle

$$a=b=c$$ $$α=β=γ=60°$$ $$h={a√3}/2$$ $$r=R/2$$ $$h=r+R$$ $$h_a≡m_a≡l_a$$ $$h_b≡m_b≡l_b$$ $$h_c≡m_c≡l_c$$

Right triangle

$γ=90°$
$a,b$ - legs
$c$ - hypotenuse

$$h^2=a_1b_1$$ $$a^2=ca_1$$ $$b^2=cb_1$$

Pythagorean theorem

$$c^2=a^2+b^2$$

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If $a=b$ then:

$$α=β=45°$$ $$h=c/2$$

If α=30° then $a=c/2$