Triangle

Acute triangle

— all angles of the triangle are acute.

Obtuse triangle

— one of the angles of the triangle is obtuse (greater than 90°).

Right triangle

— one of the angles of the triangle is right (equal to 90°).

Scalene triangle

— all three sides are unequal.

Isosceles triangle

— two sides are equal.

Equilateral (regular) triangle

— all three sides are equal.

The sum of the angles of a triangle is 180°

$\alpha +\beta +\gamma =\mathrm{180°}$

In a triangle, the greater angle lies opposite the greater side, and vice versa. Equal angles lie opposite equal sides

• if $\alpha >\beta$, then $a>b$
• if $\alpha =\beta$, then $a=b$

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

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

Law of sines

The sides of a triangle are proportional to the sines of the opposite angles.

$\frac{a}{sin\alpha }=\frac{b}{sin\beta }=\frac{c}{sin\gamma }$

Law of cosines

The square of any side of a triangle equals the sum of the squares of the other two sides minus twice the product of these sides and the cosine of the angle between them.

$a^2=b^2+c^2-2bc·cos\alpha$
$b^2=a^2+c^2-2ac·cos\beta$
$c^2=a^2+b^2-2ab·cos\gamma$

Projection theorem

For an acute triangle:

$a=bcos\gamma +ccos\beta$
$b=acos\gamma +ccos\mathrm{\alpha ;}$
$c=acos\beta +bcos\mathrm{\alpha ;}$

Formulas for the sides in terms of the medians

$a=\frac{2}{3}\sqrt{2\left({m_b}^2,+,{m_c}^2\right)-{m_a}^2}$

$b=\frac{2}{3}\sqrt{2\left({m_a}^2,+,{m_c}^2\right)-{m_b}^2}$

$c=\frac{2}{3}\sqrt{2\left({m_a}^2,+,{m_b}^2\right)-{m_c}^2}$

Formula for the area of a triangle by side and height

The area of a triangle equals half the product of the length of a side of the triangle and the length of the height drawn to that side.

$S=\frac{1}{2}a·h_a$ ,
$S=\frac{1}{2}b·h_b$ ,
$S=\frac{1}{2}c·h_c$ ,

where a, b, c are the sides of the triangle,
h_a, h_b, h_c are the altitudes drawn to the sides a, b, c of the triangle.

Formula for the area of a triangle by three sides

Heron's formula is a formula for computing the area of a triangle S from the lengths of its sides a, b, c.

$S=\sqrt{p\left(p,-,a\right)\left(p,-,b\right)\left(p,-,c\right)}$,

where p is the semiperimeter of the triangle: $p=\frac{a+b+c}{2}$
a, b, c are the sides of the triangle.

Formula for the area of a triangle by two sides and the angle between them

The area of a triangle equals half the product of two of its sides multiplied by the sine of the angle between them.

$S=\frac{1}{2}a·b·sin\gamma$ ,
$S=\frac{1}{2}b·c·sin\alpha$ ,
$S=\frac{1}{2}a·c·sin\beta$ ,

where a, b, c are the sides of the triangle,
γ is the angle between the sides a and b,
α is the angle between the sides b and c,
β is the angle between the sides a and c.

Formula for the area of a triangle by three sides and the radius of the circumscribed circle

$S=\frac{a·b·c}{4R}$ ,

a, b, c are the sides of the triangle,
R is the radius of the circumscribed circle.

Formula for the area of a triangle by three sides and the radius of the inscribed circle

The area of a triangle equals the product of the semiperimeter of the triangle and the radius of the inscribed circle.

$S=p·r$ ,

where S is the area of the triangle,
r is the radius of the inscribed circle,
p is the semiperimeter of the triangle: $p=\frac{a+b+c}{2}$

By two sides and the angle between them

Theorem.

If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.

By a side and two adjacent angles

Theorem.

If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.

By three sides

Theorem.

If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

Definition

Similar triangles are triangles in which the angles are respectively equal and the sides of one are respectively proportional to the sides of the other triangle.

$\mathrm{∆АВС}~\mathrm{∆MNK}=>\alpha ={\alpha }_1$,$\beta ={\beta }_1$,$\gamma ={\gamma }_1$ и $\frac{\mathrm{AB}}{\mathrm{MN}}=\frac{\mathrm{BC}}{\mathrm{NK}}=\frac{\mathrm{AC}}{\mathrm{MK}}=k$

where k is the similarity coefficient.

Criteria for similarity of triangles

1. If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar.
2. If three sides of one triangle are proportional to three sides of another, then such triangles are similar.
3. If two sides of one triangle are proportional to two sides of another, and the angles between these sides are equal, then such triangles are similar.

Properties

The areas of similar triangles are related as the square of the similarity coefficient:

$\frac{S_{\mathrm{∆АВС}}}{S_{\mathrm{∆MNK}}}=k^2$

Properties of a right triangle

• The sum of the two acute angles of a right triangle is 90°.
  The sum of the angles of a triangle is 180°, and the right angle is 90°, therefore the sum of the two acute angles of a right triangle is $\mathrm{\angle \; 1}+\mathrm{\angle \; 2}=\mathrm{90°}$.
• 

  The leg of a right triangle lying opposite an angle of 30° equals half the hypotenuse (the hypotenuse is twice as long as the leg lying opposite an angle of 30°).

  Consider a right triangle ABC, in which ∠ A is right, ∠ B = 30°, and hence ∠ C = 60°.

  Let us prove that BC=2AC.
  Attach to triangle ABC a triangle ABD equal to it, as shown in the figure.
  We obtain triangle BCD, in which ∠ B = ∠ D = 60°, therefore DC = BC. But DC = 2AC. Consequently, BC = 2AC.

  The converse is also true: If a leg of a right triangle equals half the hypotenuse (or the hypotenuse is twice as long as the leg), then the angle lying opposite this leg is 30°.

Criteria for congruence of right triangles

Since in a right triangle the angle between the two legs is right, and any two right angles are equal, from the general criteria for congruence of triangles one can formulate specific criteria for congruence of right triangles.

1. If the legs of one right triangle are respectively equal to the legs of another, then such triangles are congruent.
2. If a leg and the acute angle adjacent to it of one right triangle are respectively equal to a leg and the acute angle adjacent to it of another, then such triangles are congruent.
3. If the hypotenuse and an acute angle of one right triangle are respectively equal to the hypotenuse and an acute angle of another, then such triangles are congruent.
4. If the hypotenuse and a leg of one right triangle are respectively equal to the hypotenuse and a leg of another, then such triangles are congruent.

Properties

The areas of similar triangles are related as the square of the similarity coefficient:

$\frac{S_{\mathrm{∆АВС}}}{S_{\mathrm{∆MNK}}}=k^2$

See also: Pythagorean theorem.