
Plane Geometry, Plane Figures (Geometric
Figures)  Triangles 


Oblique
or Scalene Triangle 
Properties and rules

Perpendicular bisectors,
triangles circumcenter

Angle bisectors, the center of the triangle’s incircle

The
median, the centroid of a triangle 
The altitude of a triangle, orthocenter

Triangle formulas 
Similarity and congruence of
triangles use

Congruence 





Properties and rules

The sum of the angles of a triangle
is a +
b
+ g
= 180°. 
Any side of a triangle is shorter than the sum of other two
sides. 
Circumcircle 
A
perpendicular bisector of a triangle is a straight line
passing through the midpoint of a side and being perpendicular
to it. 
The
perpendicular bisectors of the 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. 



Incircle 
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. 



Median
and 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. 



Altitude
and orthocenter 
The
altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. 
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
semiperimeter. 




Similarity and congruence of
triangles use

Similarity

Two triangles (or two plane figures) are similar if they have
corresponding angles equal 
a = a', b
= b', g
= g', 
hence corresponding pairs of sides in
proportion. 
If
k is the ratio of sides of two similar triangles, then
a'
= ka,
b'
= kb,
c'
= kc, 
hence 
P'
= kP 
and 
A'
= k^{2}A 




Example:
Given triangle ABC
is divided by the angle a
bisector into two triangles ABD
and ADC,
as is shown in the picture. 


By
use of the similarity it can be shown that 

Proof:
Through the point D
drawn is the line segment DE
parallel to the side c,
hence the triangles, ABC
and EDC
are similar. Therefore 


Example:
Find the value of x
of the triangle shown in the picture. 

Solution: 





Congruence

Two triangles are congruent if they have identical size and shape
so that they can be exactly superimposed. 
Thus, two triangles are
congruent: 
a) if a pair of corresponding sides and the included
angle are equal, 
b) if their corresponding sides are
equal, 
c) if a
pair of corresponding angles and the included side are equal. 

The
sine law
(or the sine rule) and the cosine law 
From the congruence of triangles follows that an oblique triangle is determined 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 determine 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 an oblique triangle. 








Geometry
and use of trigonometry contents  A 



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