(

*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.