Notion of Triangle and Different Types of Triangles
Before discussing different types of triangles, let’s learn what the triangle is. Triangle in Euclidean space is the geometric figure formed by three segments that connect the three points that do not lie on a straight line. These three points are called vertices of the triangle, and the segments are the sides of the triangle. The sides of the triangle form three angles in the vertices of the triangle. In other words, triangle is a polygon, which has exactly three angles. If the three points lie on one straight line, then the triangle with vertices at these three points is called degenerate. All other triangles are non-degenerate triangles.
In the non-Euclidean spaces the geodesic lines act as the sides of the triangle, which are typically curved. Therefore, such triangles are called curvilinear. An important special case of non-Euclidean triangles are spherical triangles.
Elements of a Triangle
The basic elements of any triangle are: vertices, sides, and angles.
Now let’s talk about how to correctly identify and name them.
- The vertices of the triangle are nothing else than the usual points in the vertices of the angles that form a triangle. You know that points in the plane are indicated by capital letters: A, B, C, D, etc. So when you are asked to specify the vertices of the triangle, you just need to write the names of these points.
- The sides of the triangle are segments that form it. The segment is a part of the line, bounded by two points (the segment ends). The segments are marked according to the names of their ends, i.e., with a pair of large letters, for example: AB, BC, CD, etc. Note that you can name the same segment as AB or BA – there is no difference.
- The inner angles of a triangle can be called by its vertices, but before marking each corner you need to put a special character in front of it. The angles of a triangle can be written in a different way as well. Every corner can be marked with three points, but you must remember that the apex of the angle should always be in the middle.
Qualities of Diverse Types of Triangles
- External angle is equal to the difference between 180° and an internal angle, which can take values from 0 to 180°.
- The Theorem of the external angle of the triangle is true to the outside angle of a triangle: the outer angle of the triangle is equal to the sum of the other two interior angles, which is not related to it.
Signs of Equality of Triangles
The triangle in the Euclidean plane is uniquely (within the congruence) can be determined according to the following three key elements:
- a, b, y (equality according to two sides and the angle between them);
- a, b, y (equality according to one side and an adjacent two angles);
- a, b, y (equality according to the three sides).
The equality of right triangles can be determined by the following signs:
- according to the leg and a hypotenuse;
- according to two cathetuses;
- according to a leg and an acute angle;
- according to the hypotenuse and an acute angle.
In spherical geometry, there is an indication of equality of triangles by three corners.
- Types of Triangles
The types of tringles can be defined by the magnitude of the angles and the number of equal sides.
- Types of Triangles by the Magnitude of the Angles
Since in geometry Euclidean the sum of the corners of a triangle equals 180°, then at least two corners of the triangle should be acute (less than 90°). So, by the magnitude of the angles there are the following types of triangles:
- If all the angles of a triangle are acute, then this triangle is called an acute-angled;
- If one of the angles of a triangle is obtuse (more than 90°), then the triangle is called obtuse;
- If one of the angles of a triangle is right (equal to 90°), then the triangle is called rectangular. The two sides forming the right angle are called legs, while the side opposite the right angle is called the hypotenuse.
In hyperbolic geometry the sum of angles of a triangle is always less than 180°, while it is always more in the sphere. The difference between the sum of the angles of a triangle and 180° is called the defect. The defect is proportional to the area of a triangle, thus, the sum of the angles of infinitely small triangles on the sphere or hyperbolic plane won’t differ much from 180°.
Types of Triangles by the Number of Equal Sides
- Triangle with all three unequal sides is called versatile (equilateral).
- Isosceles triangle is a triangle with two equal sides. These sides are called lateral, while the third side is called the base. In an equilateral triangle, the angles at the base are equal. Height, the median, and the bisector of an isosceles triangle, put on the base of a triangle coincide.
- Equilateral or right triangle is a triangle with all three equal sides. In an equilateral triangle all the angles are 60°, and the centers of the inscribed and circumscribed circles coincide.
Definitions Related to Various Types of Triangles
All the facts set out in this section are from the Euclidean geometry.
The perimeter of different types of triangles is the sum of the lengths of its three sides, and half of this quantity is called semiperimeter.
Rays, Segments and Points of Types of Triangles
The median in all the types of triangles, conducted from a given vertex, is the segment connecting the vertex with the middle of the opposite side (the basis of the median). The three medians of a triangle intersect at one point. This point of intersection is called the centroid or center of gravity of the triangle. The last name is due to the fact that the triangle made of homogeneous material has the center of gravity in the point of intersection of the medians. Centroid divides each in median according to the ratio of 1:2, counting from the base of the median. The triangle with vertices at the midpoints of the medians is called the median triangle.
The bases of the medians of all types of triangles form a so-called additional triangle.
The height in all types of triangles, conducted from a given vertex, is the perpendicular dropped from the vertex to the opposite side or its continuation. Three heights of a triangle intersect at one point, which is called the orthocenter of the triangle. The triangle with vertices in the bases’ of heights is called orthocentric.
Bisector in different types of triangles, held out of the top, is the segment connecting the vertex with the point on the opposite side and dividing the angle of the top half. The bisectors of a triangle intersect at one point, and this point coincides with the center of the inscribed circle.
The segment in types of triangles joining a vertex with a point on the opposite side is called Ceva line. Usually, Ceva line doesn’t imply one such segment, but one of the three such line segments drawn from three different vertices of the triangle and intersect at one point. They satisfy the conditions of Ceva’s theorem.
The central line in various types of triangles is the line connecting the middles of the two sides of this triangle. The three middle lines of the triangle divide it into four equal triangles with an area less in 4 times, than the original area of the triangle.
Midperpendiculars to the sides of diverse types of triangles also intersect at one point, which coincides with the center of the circumscribed circle.
In an equilateral triangle, the median, the height, and the bisector, conducted to the base, coincide. The opposite is true: if the bisector, the median, and the height, conducted from one vertex, coincide, then the triangle is isosceles.
If the triangle is versatile, then the inner bisector, conducted from any vertex, lies between the inner median and the height conducted from the same tops vertices.
Ceva lines lying on the straight line that isotomically correspondent to the bisectors that are relevant to the bases of medians, are called symmedians. They pass through a single point – Symmedian point.
Ceva lines lying on the straight lines that isotomically correspondent to the bisectors that are relevant to the bases of medians, are called antibisectors. They pass through one point – the center of antibisectors.
Cleaver in all types of triangles is the segment, one vertex of which is in the middle of one side of the triangle, while the other vertex is located on one of the two remaining sides. Cleaver divides the perimeter in half.
Some of the points in various types of triangles form a pair. For example, there are two points, from which all sides are visible at an angle of 60°, or at an angle of 120°. These points are called Torricelli points. Also, there are two points, the projections of which on the sides lie at the vertices of an equilateral triangle. This are points of Apollo. Points P and Q are such that angle ABP = angle BCP = angle CAP, and angle BAP = angle CBP = angle ACP are called Brocard points.
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