(
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 lC is a line that is passing through
vertex C and is
perpendicular to
lAB, the line that passes through
A and
B.
C
/|\
/ a \
/ l \
/ t \
/ i \
/ t \
/ u \
/ d \
/ e \
A-----------------+---------B
Fig. 1 - Showing altitude lC.
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 lC and three other triangles congruent to ABC.
Altitude
lC 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.