### Triangle Incircle Iteration

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this process starting with T2 to form a sequence of nested triangles and circles. What happens to the triangles? You may like to investigate this interactively on the computer or by drawing with ruler and compasses. If the angles in the first triangle are a, b and c prove that the angles in the second triangle are given (in degrees) by f(x) = (90 - x/2) where x takes the values a, b and c. Choose some triangles, investigate this iteration numerically and try to give reasons for what happens. Investigate what happens if you reverse this process (triangle to circumcircle to triangle...)

### First Forward Into Logo 1: Square Five

A Short introduction to using Logo. This is the first in a twelve part series.

### First Forward Into Logo 10: Count up - Count Down

What happens when a procedure calls itself?

# Stringing it Out

##### Stage: 4 Challenge Level:

The invitation is for you to explore the following transformations and comment on what you find:

### First

T(x) = y
T(y) = xy

This transformation has the following successive effect (starting with x):

x
y
xy
yxy
xyyxy
...
...

### Next

R(x) = xy
R(y) = z
R(z) = xy

This transformation has the following successive effect:

x
xy
xyz
xyzxy
xyzxyxyz
...
...

### Then

S(w) = wx
S(x) = y
S(y) = wz
S(z) = y

This transformation has the following successive effect:

w
wx
wxy
wxywz
...
...