
Plane Figures 
Triangle 


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 





Oblique
or Scalene Triangle 
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.









Beginning
Algebra Contents D 



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