Assignment 6
Problem for GSP
by Emily Bradley
Given three points A, B, and C. Draw a line intersecting AB in the point X and BC in the point Y such that
AI = IH = HC
Begin by looking at the case where segments AB and BC are equal and angle ABC is right. The bright blue segments in the diagram are all of equal lengths. Forming a parallelogram will allow us to make the figure in which AI=IH=HC. But how do we form the parallelogram for any three lines. In the case of the 454590 triangle. Ray AH is the angle bisector to angle BAC and gives us the length of segment HC. This will not be the case for triangles that are not isosceles. But how do we find this parallelogram?
Notice the smaller similar parallelogram. This was formed by an arbitrary point chosen on segment AB. We can work backwards by creating this parallelogram for any three points and we are then able to find the larger similar parallelogram.
The following GSP sketch allows one to move the vertexes and see this hold true in all cases.
Steps to Completion
Beginning with the three points A, B, C, construct the angle <ABC 


We connect AC as is completing the triangle ABC. 

Choose any point D on the segment AB. 

Construct a circle with center D. 

Construct a line through D parallel to segment BC. 

Label the intersection of this line and the circle E. Construct a new circle with center E. 

Label the intersection of the circle and segment AC, F. Connect EF. 

Draw a line through F parallel to BC and DE. Draw a line through D parallel to EF. 

These segments are all radii and have the same length, therefore parallelogram DEFG has been constructed. 

Draw ray AG. 

Where AG intersects segment BG, call this point H. Construct circle with center H. 

Where circle H intersects segment BA, call this point I. Now CH and CI are both radii. Circle with center I, radius IH, has radius IA as well. 

And AI, IH, HC are all the same length. 