## Why does this work j It works because, if we call the top point of the triangle T, then BT is half

Ias long as AB. So suppose we say AB has length 1. Then BT will have length 1/2. We can find the length of the other side of the triangle, the diagonal AT, by using Pythagoras' Theorem:

AT2 = 1 + 1/4 = 5/4 Now, taking the square-root of each side gives:

Point V was drawn so that TV is the same length as TB = AB/2 = 1/2. So AV is just AT - TV = (V5)/2 -1/2 = phi.

The final construction is to mark a point G which is same distance (AV) along the original line (AB) which we do using the compasses.

So AG is phi times as long as AB!

1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

This time we find a point outside of our line segment AB so that the new point defines a line which is Phi (1-618..) times as long as the original one.

Here's how to find the new line Phi times as long as the original: 1. First repeat the steps 1 and 2 above so that we have found the mid-point of AB and also have a line at right angles at point B.

2. Now place the compass point on B and open them out to touch A so that you can mark a point T on the vertical line which is as long as the original line.