Basic Elements of a Triangle
The basic elements of a triangle are:
- Vertices: points where two sides meet. It has 3 vertices (A, B, and C).
- Sides: line segments that join two consecutive vertices of the triangle and that delimit its perimeter. It has 3 sides (a, b, and c).
- Interior angles: angles that form two consecutive sides at the vertex where they converge. There are 3 interior angles (α, β and γ). The interior angles of the triangle add up to 180° (why do they add up to 180°?):
- Exterior angles: angle of one side with the exterior extension of the consecutive side. There are 3 exterior angles (θ). The outside angles always add up to 360°.
- Altitude of a Triangle: the altitude (or height) of a triangle (h) is a line segment perpendicular to a side that goes from the opposite vertex to this side (or its extension). It can also be understood as the distance from one side to the opposite vertex. A triangle has three heights, depending on the reference vertex that is chosen. The three altitudes intersect at a point called the orthocenter.
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Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
In a right triangle, different elements can be distinguished, referring to its sides and angles.
- Legs (or cathetus): are the sides of the triangle that together form the right angle.
- Hypotenuse: is the largest side of the triangle opposite the right angle.
- Right angle: is a 90° angle formed by the two legs.
- Acute angles: the other two angles of the triangle (α and β) are less than 90°.
Every triangle has lines (also called cevians) and points that determine a number of important elements.
There are many different constructions that find a special point associated with a triangle, satisfying some unique property. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point.
The altitude of a triangle, or height, is a line from a vertex to the opposite side, that is perpendicular to that side. It can also be understood as the distance from one side to the opposite vertex.
Every triangle has three altitudes (ha, hb and hc), each one associated with one of its three sides. If we know the three sides (a, b, and c) it’s easy to find the three altitudes, using the Heron’s formula:
- Obtuse triangle: The altitude related to the longest side is inside the triangle (see hc, in the triangle above) the other two heights are outside the triangle (ha, and hb).
- Right triangle: The altitude with respect to the hypotenuse is interior, and the other two altitudes coincide with the legs of the triangle.
- Acute triangle: all three altitudes lie inside the triangle.
Where is the orthocenter located?
- Obtuse triangle: the orthocenter is outside the triangle.
- Right triangle: the orthocenter coincides with the right angle’s vertex.
- Acute triangle: the orthocenter is an inner point.
Where a, b, and c are the sides of the triangle with respective medians ma, mb and mc from their midpoints.
In any median of a triangle, the distance between the center of gravity (or centroid) G and the center of its corresponding side is one third (1/3) of the length of that median, i.e., the centroid is two thirds (2/3) of the distance from each vertex to the midpoint of the opposite side.
The perpendicular bisector of a segment is a line perpendicular to the segment that passes through its midpoint. It has the property that each of its points is equidistant from the segment’s endpoints.
The radius (R) of the circumcircle is given by the formula:
There are three angle bisectors (Ba, Bb and Bc), depending on the angle at which it starts. We can find the length of the angle bisector by using this formula:
The radius (or inradius) of the inscribed circle can be found by using the formula:
The main points and lines (cevian) associated with a triangle are summarized in the following list::
AUTHOR: Bernat Requena Serra