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Geometry and use of trigonometry
 Plane figures
 Triangle
  Types of triangles classified by their sides are; a scalene triangle, isosceles triangle and equilateral triangle, or when classified by angles as; an acute triangle, obtuse triangle and right triangle.
Properties of triangles
a)  A sum of a triangle angles a + b + g = 180°. 
b)  Angles, lying opposite the equal sides, are also equal, and inversely.
c)  In any triangle, if one side is extended, the exterior angle is equal to a sum of interior angles not supplementary.
d The sum of exterior angles is 360°.

e)  Any side of a triangle is less than a sum of two other sides and greater than their difference.

Proof: By turning the side BC, of the scalene triangle ABC above, around the vertex C by the angle g into direction of the side AC, obtained is the isosceles triangle BCD with equal angles on the base BD.

Its lateral side CD < AC. Angle a < b', as b' is the exterior angle of the triangle ABD, and b' < b, therefore a < b. Thus, proved is the above statement.

Congruence of triangles
Two figures are called congruent if they have identical size and shape, i.e., if their corresponding angles and sides are equal.
The two congruent figures fit on top of each other exactly. We prove the congruence of two figures by rotation and translation.
Theorems about congruence of triangles are; Two triangles are congruent:
a)  If a pair of corresponding sides and the included angle are equal SAS (Side-Angle-Side).
b)  If their corresponding sides are equal SSS. 
c)  If a pair of corresponding angles and the included side are equal ASA.
The congruence of two triangles we denote as D ABC @ D A'B'C'.
Similarity of triangles

Two plane figures are similar if differ in scale not in shape. Two polygons are similar, if their angles are equal and sides are proportional.

Similarity criteria of triangles  -  Two triangles are similar:
a)  If all their corresponding angles are equal.
b)  If all their sides are proportional.
c)  If one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional.
Similarity of two triangles is denoted as D ABC ~ D A'B'C'.
Right-angled Triangle
The Pythagorean theorem
In any right triangle the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares on the other two sides (legs).
From the similarity of the triangles, ADC, BDC and ABC, and Thales’ theorem (an angle inscribed in a semicircle is a right angle) proved is Pythagoras’ theorem,
Trigonometric functions of an acute angle defined in a right triangle
Trigonometric functions of an acute angle are defined in a right triangle as a ratio of its sides.
Solving the right triangle
To solve a right triangle means to find all unknown sides and angles using its known parts.
While solving a right triangle we use Pythagoras’ theorem and trigonometric functions of an acute angle depending which pair of its elements is given.

Note, right triangles are usually denoted as follows;  c stands for the hypotenuse,  a and b for the perpendicular sides called legs, and a and b for the angles opposite to a and b respectively.

Oblique or scalene triangle
The triangle's circumcircle, perpendicular bisectors

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it. Perpendicular bisectors of sides of any triangle are concurrent (all pass through the same point). The perpendicular bisectors intersect in the triangle's circumcenter.

The triangle's circumcenter is the center of the circumcircle which circumscribes given triangle passing through all its vertices.
Acute triangles' circumcenter falls inside the triangle. Obtuse triangles' circumcenter falls outside the triangle.
The triangle's incircle, the angle bisector
An angle bisector is a straight line through a vertex of a triangle that divides the angle into two equal parts.
The three angle bisectors intersect in a single point called the incenter, the center of the triangle's incircle.
Incircle is a circle inscribed in a triangle so that each of the sides of the triangle is a tangent, of which the radius is inradius, therefore the radius is perpendicular from the incenter to any side.
The median of triangle, centroid
The median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.
The medians intersect at the triangle's centroid.
The centroid cuts every median in the ratio 2 : 1 from a vertex to the midpoint of the opposite side.
The altitude of triangle, orthocenter
A height or altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude.
The altitudes intersect in the orthocenter of the triangle. See the picture above.
Triangle formulas
Meaning of symbols used in the above pictures and in the triangle formulas are:
h -altitude,  m -median,  t -angle bisector,  r -radius of the incircle,  R -radius of the circumcircle,  A -area,
P -perimeter,  s -semi-perimeter.
The area A of any triangle is equal to one-half the product of any base and corresponding height h.
The sine law (or the sine rule) and the cosine law
From the congruence of triangles follows that an oblique triangle is defined by three of its parts, as are
     - two sides and the included angle (SAS),    - two angles and the included side (ASA),
     - three sides (SSS)                 and              - two sides and the angle opposite one of them (SSA),
which does not always define a unique triangle.
By using definitions of trigonometric functions of an acute angle and Pythagoras’ theorem, we can examine mutual relationships of sides and corresponding angles of a (an oblique) triangle.
 The sine law
From the right triangles, BCD and ACD in the figure,
hc = a sinb   and   hc = b sina,
so that,            a sinb = b sina,

Expressing the same way the altitudes, hb and ha as common legs of another pairs of two right triangles in the above triangle, we get

hb = a sing   and   hb = c sina,
so that,            a sing = c sina,
and,   ha = b sing   and   ha = c sinb,
      so that,               b sing = c sinb,
These relations are called the sine law and in words: Sides of a triangle are to one another in the same ratio as sine of the corresponding (opposite) angles.
 The cosine law
From the right triangles, BCD and ABD in the figure,

Expressing the same way the squares of the altitudes as common legs of another two pairs of the right triangles, obtained are

 
 
 
 
 
 
 
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