**Exploration 6:**

**AX = XY = YB**

John Watson

Let's say you have two segments AC and BC that intersect at a point C. Construct a line segment intersecting AC in the point X and BC in the point Y such that...

AX = XY = YB

The following GSP construction satisfies these conditions...

Proof:(i) The red circle, centered at X, passes through point A and point Y.

(ii) Therefore, length of segment AX = length of segment XY because they are both radii of the same circle.

(iii) The blue circle, centered at Y, passes through point X and point B.

(iv) Therefore, length of segment YB = length of segment XY because they are both radii of the same circle.

(v) By transitivity we have length of segment AX = length of segment XY = length of segment YB

I have created a script tool that can be found here for making such constructions.

Open up the script view to see expicit instructions for how to create the construction.