Plane Geometry, Plane Figures (Geometric Figures) - Triangles
Oblique or Scalene Triangle   Angle bisectors, the center of the triangle’s incircle The median, the centroid of a triangle The altitude of a triangle, orthocenter  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 -semi-perimeter.  where     where 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' = kab' = kbc' = kc,
 hence P' = kP and A' = k2A 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 similarly 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 