Points, Lines, and Circles Associated with a Triangle
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.
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Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
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::
- Median………….. Centroid (G)
- Altitude……………… Orthocenter (H)
- Perpendicular Bisector……….. Circumcenter (O)
- Angle Bisector…………. Incenter (I)
Find the lengths of the three altitudes, ha, hb and hc, of the triangle Δ ABC, if you know the lengths of the three sides: a=3 cm, b=4 cm and c=4.5 cm.
Firstly, we calculate the semiperimeter (s).
We get that semiperimeter is s = 5.75 cm. Then we can find the altitudes:
The lengths of three altitudes will be ha=3.92 cm, hb=2.94 cm and hc=2.61 cm.
Find the length of the median of a triangle Δ ABC if length of sides are a=2 cm, b=4 cm and c=3 cm.
Using the equation of Apollonius’ theorem, we can find the length of all three medians:
Thus, the medians are ma=3.39 cm, mb=1.58 cm and mc=2.78 cm.
Find the radius R of the circumscribed circle (or circumcircle) of a triangle of sides a = 9 cm, b = 7 cm and c = 6 cm.
We get the semiperimeter s:
We apply the formula for the radius R of the circumscribed circle, giving the following values:
So, the radius R is 4.5 cm.
Let ABC de a triangle in which a=3 cm, b=4 cm and c=2 cm. Find the angle bisectors Ba, Bb y Bc.
We’re going to solve this exercise using the angle bisector’s formula:
Since we already know what all the sides of the triangle measure, we only have to find the semiperimeter (s):
Then, we can substitute the values in the angle bisector’s formula:
Therefore, Ba=2.45 cm, Bb=1.47 cm and Bc=3.32 cm.
AUTHOR: Bernat Requena Serra