Equal Segments
by
Susan Sexton
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 = YB.
To begin, I thought it was a
simple matter of using circles. I
picked a point, X, on AC and constructed a circle whose center is at X and
whose radius is AX.
Next I constructed a circle
whose center is at B and whose radius is AX.
Then I enlarged circle X
until the circles intersected on segment CB at some point Y.
Since XY is a radius of circle
X then it is equal to AX and since circle B was constructed to be congruent to
circle X then YB must be equal to XY and AX. So then it is complete.
But that works when you have
a nice drawing software package such as GSP.
How would I find these segments without moving the circles around? Back to square one.
At least I knew that if I
constructed AB then I would have quadrilateral AXYB. So perhaps I can recreate the quadrilateral or at least a
similar quadrilateral.
So I again picked a point,
now called X',
on segment AC and constructed a circle whose center is at X' and whose radius is AX'. Next I
constructed a circle whose center is at B and whose radius is AX'. I
labeled the intersection of this circle with segment BC as Y'. Next I
constructed AB.
Then I attempted to construct
a quadrilateral similar to my desired quadrilateral off of segment AX'. So I
constructed a line parallel to AB through Y' that intersects circle X' at point Y''. Next I
constructed a line parallel to CB through Y''. This line intersects AB at point B'.
So AX' = X'Y'' = Y''B' and I have a similar, smaller version of the desired
segments.
Now I will clean up the
sketch a bit to focus on just the similar figure.
So if I construct a line
through A and Y'' then that line will intersect segment BC at the true Y.
Next if I construct a line
parallel to X'Y'' through Y then
that line will intersect AC at the true X.
So I have constructed the
desired segments and I can confirm this with the measurement tools.
Here is a script of the finding AX = XY = YB for any length AC and CB.