Hello, gwen01!

Divide a line segment into five equal parts using a compass and straightedge.

Given a line segment AB diagonally (say, to the upper-right).

Draw a horizontal line AC to the right. Code:

B
*
*
*
*
*
*
*
*
*
*
A * - - - - - - - - - - - - - - - - - C

On AC, mark off five equal segments: . Code:

B
*
*
*
*
*
*
*
*
*
*
A * - - + - - + - - + - - + - - + - - C
P Q R S T

Draw segment BT. Code:

B
*
* \
* \
* \
* \
* \
* \
* \
* \
* \
A * - - + - - + - - + - - + - - + - - C
P Q R S T

Through P, Q, R, S construct lines parallel to BT,

. . intersecting AB at W, X, Y, Z. Code:

B
*
Z * \
o \
Y * \ \
o \ \
X * \ \ \
o \ \ \
W * \ \ \ \
o \ \ \ \
* \ \ \ \ \
A * - - + - - + - - + - - + - - + - - C
P Q R S T

Then: .

Nice job, wingless!