|Figure 1: Examples of triangles
|Figure 2: Examples of shapes that are not triangles.
Parts of a Triangle
Types of Triangles
Properties of Triangles
Perimeter of a Triangle
Angle Sum Theorem
Area of a Triangle
Heron's Formula for Area of a Triangle
Incircle and Incenter of a Triangle
Circumcircle and Circumcenter of a Triangle
Median of a Triangle
Centroid of a Triangle
Altitude of a Triangle
Orthocenter of a Triangle
Euclid. Elements, Book 1 Proposition 6: If two sides of a triangle are equal, the angles opposite the equal sides are equal.
Centers of a Triangle
A triangle has three angles, three vertices, three sides and three pairs of exterior angles.
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|A triangle with one right angle.
|A triangle with three acute angles.
|A triangle with one obtuse angle.
|A triangle whose sides are all different lengths.
|A triangle with three equal sides.
|A triangle with two equal sides.
|Figure 3: Types of triangles
|By convention, triangles are usually labeled in a counterclockwise direction, often using the letters A, B, and C. The sides are often labeled with a lower case letter corresponding to the vertex opposite the side.
The perimeter of a triangle is the sides of the triangle or the sum of the lengths of the sides. For example, if the lengths of the sides are 3, 4, and 5, the perimeter is 3 + 4 + 5 = 12.
|Angle Sum Theorem
|In Euclidean geometry, the sum of the angles of a triangle is 180° = π radians. In other geometries, this might not be true.
The area of a triangle is where b (base) is any side of the triangle, and h (height) is the distance from the vertex opposite the base (in this case B) to the extended base (in this case the line AC).
The area of a triangle can also be calculated from the length of the three sides using Heron's Formula. First, one must calculate the semiperimeter. This 1/2 of the perimeter. Since the perimeter is a + b + c where a, b and c are the length of the sides of the triangle, the semiperimeter is .Heron's formula for the area of a triangle is
The incircle of a triangle is the circle that is tangent to each of the sides of a triangle. The incenter is the center of the incircle. For more information on the incenter of a triangle, see Incenter.
|The circumcircle of a triangle is the circle that passes through all of the vertices of a triangle. The circumcenter is the center of the circumcircle. For more information on the circumcenter or circumcircle of a triangle, see Circumcenter from All Math Words Encyclopedia.
A median of a triangle is a line drawn through a vertex of the triangle and the midpoint of the opposite side. This means that every triangle has three medians. The medians of a triangle meet at a point called the centroid of the triangle.
The centroid of a triangle is the center of gravity of the triangle. This means that if a triangle is balanced on a pin at the centroid, it would be perfectly balanced.
|An altitude of a triangle is a line segment from a vertex of the triangle to the extended opposite side, perpendicular to the opposite side.
|The orthocenter of a triangle is at the intersection of the altitudes of a triangle.
Two triangles are congruent if two adjacent sides and the angle contained by the sides are congruent with corresponding sides and angle of the other triangle. In this case we say that the triangles are SAS congruent. SAS stands for side, angle, side.
For more information on SAS Congruence, see SAS Congruence.
|Proposition 6, Euclid's Elements: If two angles of a triangle are equal, the sides opposite the equal angles are also equal.
In a triangle, if two angles have equal length, the sides opposite the equal angles are also equal. In figure 16, the angle ABC is equal to the angle ACB. The side AB is also equal to the side AC.
For more information on this property of triangles see:
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For what class of triangle are the centroid, orthocenter and circumcenter coincidental?
|Manipulative 13 - Centers of a Triangle Created with GeoGebra.
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