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Geometry: Theorems

Theorems for Segments within Triangles

Problems

Problems

In this lesson we'll learn properties of altitudes, medians, midsegments, angle bisectors, and perpendicular bisectors of triangles. All four of these types of lines or line segments within triangles are concurrent, meaning that the three medians of a triangle share intersecting points, as do the three altitudes, midsegments, angle bisectors, and perpendicular bisectors. The intersecting point is called the point of concurrency. The various points of concurrency for these four types of lines or line segments all have special properties.

Altitudes of a Triangle

The lines containing the altitudes of a triangle meet at one point called the orthocenter of the triangle. Because the orthocenter lies on the lines containing all three altitudes of a triangle, the segments joining the orthocenter to each side are perpendicular to the side. Keep in mind that the altitudes themselves aren't necessarily concurrent; the lines that contain the altitudes are the only guarantee. This means that the orthocenter isn't necessarily in the interior of the triangle.

Figure %: The lines containing the altitudes of a triangle and the orthocenter

There are two other common theorems concerning altitudes of a triangle. Both concern the concept of similarity. The first states that the lengths of the altitudes of similar triangles follow the same proportions as the corresponding sides of the similar triangles.

The second states that the altitude of a right triangle drawn from the right angle to the hypotenuse divides the triangle into two similar triangles. These two triangles are also similar to the original triangle. The figure below illustrates this concept.

Figure %: Triangles ABC, DAC, and DBA are similar to one another

Medians of a Triangle

Every triangle has three medians, just like it has three altitudes, angle bisectors, and perpendicular bisectors. The medians of a triangle are the segments drawn from the vertices to the midpoints of the opposite sides. The point of intersection of all three medians is called the centroid of the triangle. The centroid of a triangle is twice as far from a given vertex than it is from the midpoint to which the median from that vertex goes. For example, if a median is drawn from vertex A to midpoint M through centroid C, the length of AC is twice the length of CM. The centroid is 2/3 of the way from a given vertex to the opposite midpoint. The centroid is always on the interior of the triangle.

Figure %: A triangle's medians and centroid

Two more interesting things are true of medians. 1) The lengths of the medians of similar triangles are of the same proportion as the lengths of corresponding sides. 2) The median of a right triangle from the right angle to the hypotenuse is half the length of the hypotenuse.

Midsegments of a Triangle

The midsegment of a triangle is a segment whose endpoints are both midpoints of sides. Every triangle has three midsegments. The midsegment of a triangle is always parallel to the third side (the side whose midpoint it doesn't include), and half as long as the third side.

Angle Bisectors of Triangles

The angle bisectors of a triangle intersect each other at a point called the incircle of the triangle. The incircle of a triangle is the same as the center of a circle inscribed in a triangle. Every triangle can have exactly one inscribed circle, whose center is the incircle of the triangle, which is the point at which the angle bisectors of the triangle intersect. The incircle, then, is equidistant from the three sides of the triangle--a property that results from the inherent congruency of the radii of a circle.

Figure %: A triangle's angle bisectors and incircle

Another property of angle bisectors has to do with the side opposite the bisected angle. An angle bisector divides the side opposite the bisected angle into two segments that are of the same proportion as the other two sides. For example, in triangle ABC above, let the angle at vertex A be bisected, and let the bisector intersect BC at point D. BD/DC = BA/CA.

Perpendicular Bisectors of Triangles

The three perpendicular bisectors of a triangle intersect at one point called the circumcenter of a triangle. The circumcenter is the center of the circle circumscribed about the triangle and is equidistant from all the vertices of the triangle. In this case the perpendicular bisectors of the sides of the triangles are lines, not segments. Therefore, the circumcenter of a triangle does not necessarily exist in the interior of the triangle. Often the perpendicular bisectors of a triangle intersect outside the triangle.

Figure %: A triangle with its perpendicular bisectors and circumcenter

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