A Visual Proof of Thales's Intercept Theorem

n

1

lines

Consider an arbitrary triangle with sides

a

,

b

,

c

. Extend

b

to

b'

and

c

to

c'

to form a second triangle with third side

a'

parallel to

a

. Then the lengths of the sides satisfy the relations

a'

a

=

b'

b

=

c'

c

and hence

a'

b'

=

a

b

,

b'

c'

=

b

c

,

c'

a'

=

c

a

. In other words, the line segments that are created if two intersecting lines are intercepted by a pair of parallels are proportional.

You can drag the two red points.