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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.

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.

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.

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.