Ancient Greek mathematician
Thales of Miletus is given credit for this theorem.
My professor pronounced
Thales as "tallies" (
see footnote).
The theorem
Given
triangle ABC
and a circle
M with
AC as the diameter,
B lies on the circumference of
M if and only if the angle at
B is a right angle.
B
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
A------------------------C
The proof
Let the
origin of the plane be at the center of
M.
Then location vector
A is the
additive inverse of
C (i.e.
A = -
C).
The
dot product of vector
AB and
BC is
(A − B) · (B − C) =
(A − B) · (B + A) =
|A|² − |B|²
Suppose vectors
AB and
BC are orthogonal.
Since two vectors are orthogonal
iff their
dot product is 0, it follows
|
A|² − |
B|² = 0, which simplifies to |
A| = |
B|, so
B lies on
M.
This proves the implication.
Clearly the
converse also holds.
Footnote:
Cletus the Foetus /msg'd me and said: Most people pronounce Thales as "thay-leez," but in Greek his name was pronounced "t'-hah-lays," which might explain what your professor was trying to do.