Assignment 6

by

Allison McNeece

 Given three points A, B, and C. Draw a line intersecting AC in the point X and BC in the point Y such that: AX=XY=YC

First let's construct our three points and the line segments AB and BC.

Next pick some arbitrary point X on the segment AB. Construct a circle with center X and radius AX.

Since X is not fixed the dimensions of this circle will change depending on X.

Click below to see how the circle changes as X changes.

Next we will construct a line parallel to BC which passes through X. Label the intersection of this line with the circle as point X'

The segment X'X is a radius of Circle X so we know that AX=X'X

We will now construct a circle using X' as our center and X'X as our radius.

Note that the location of X' depends on X so the dimensions of the Circle X' will vary as X varies.

Next we will label the intersection of Circle X and BC. Call it Y. Construct the segment XY.

XY is a radius of Circle X so XY=X'X=AX

Now all we need is for these to also equal YB.

Construct a line parallel to XY which passes through X'. Label the intersection of this line with Circle X' as the point Y'.

Construct the segment X'Y'. This segment is the radius of Circle X' so it is equal to X'X which is equal to AX and XY.

A circle with Y' as the center and X'Y' as radius shows that Y'Y is also a radius.

So we have:

AX=XY=Y'Y

Now, since Y' is dependent on X', and X' is dependent on X, all we need to do is move X until Y'=C.

Do this for yourself by clicking and moving X in the figure below:

In the end we have constructed our figure such that AX=XY=YC