Theorem: (Pythagoras) If a, b, and c are the sides in a
right triangle then
a2 + b2 = c2
Proof: Let A = (x1, y1), B =
(x2, y2) and C = (x3, y3)
are the vertices of the right triangle.
By the distance
formula,
a2 = (x2 - x3
)2 + (y2 - y3 )2
and
b2 = (x1 - x3
)2 + (y1 - y3 )2
so
a2 + b2 = (x2 - x3
)2 + (y2 - y3 )2 +
(x1 - x3 )2 + (y1 -
y3 )2
= x22 - 2x2x3
+ x32 + y22 -
2y2y3 + y32 +
x12 - 2x1x3 +
x32 + y12 -
2y1y3 + y32
= x22 + x12 +
y22 + y12
+ 2x32 + 2y32
- 2x1x3 - 2y1y3 -
2x2x3 - 2y2y3
If we substitute from the Lemma, we get
= x22 + x12 +
y22 + y12
- 2x1x2 -
2y1y2
= (x2 - x1 )2 +
(y2 - y1 )2 = c2
return to
Theorem 3.4