Orthocenter

(Mathematics - Euclidean geometry)

Although the basic ideas remain unchanged since 1913, the mathematical vocabulary has changed slightly. Today, the orthocenter may be described as the intersection point of the three altitudes of a triangle.

Altitude here is defined as follows:

For triangle ABC, the altitude l_C is a line that is passing through vertex C and is perpendicular to l_AB, the line that passes through A and B.

                     C
                    /|\
                  /  a \
                /    l  \
              /      t   \
            /        i    \
          /          t     \
        /            u      \
      /              d       \
    /                e        \
   A-----------------+---------B

Fig. 1 - Showing altitude l_C.

Claim: Altitudes of a triangle are concurrent.

                  C
                 /|\
               /  | \
             /\   |  \
           /    \ |  _\
         /       _H-'  \
       /     _,-' | \   \
     /   _,-'     |   \  \
   / _,-'         |     \ \
 /,-'             |       \\
A-----------------+---------B

Fig. 2 - Showing altitudes concurrent at H.

Proof: First, construct line segments parallel to each sides of ABC to form four congruent triangles.

 B'---------------------------C---------------------------- A'
   \                         /|\                          /
    \                      /  a \                       /
     \                   /    l  \                    /
      \                /      t   \                 /
       \             /        i    \              /
        \          /          t     \           /
         \       /            u      \        /
          \    /              d       \     /
           \ /                e        \  /
            A-----------------+---------B
             \                |        /
              \               |      /
               \              |    /
                \             |  /
                 \            |/
                  \          /|
                   \       /  | 
                    \    /    :
                     \ /      
                      C'

Fig. 3 - Showing altitude l_C and three other triangles congruent to ABC.

Altitude l_C is also the perpendicular bisector of A' and B'. Since the perpendicular bisectors of a triangle are concurrent, and the perpendicular bisectors of A'B'C' are the altitudes of ABC, they are concurrent also. (See circumcenter for the proof of this last sentence.)
QED

The theorem of Snapper can also be used as another proof.

Or`tho*cen"ter (?), n. [Ortho- + center.] Geom.

That point in which the three perpendiculars let fall from the angles of a triangle upon the opposite sides, or the sides produced, mutually intersect.

 

© Webster 1913.