Exploration of Three Points

Given 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.

LetŐs construct the given
three points and angle ACB.

My first step is to construct
the segment AB. Here, I will draw it with a dotted line because it is only for
construction purposes. Also, I will identify an arbitrary point, letŐs call it
X`, on segment AC.

Now I will construct a circle
about the point X`, with a radius the length of segment X`A. Then I can
construct a line parallel to segment CB passing through X`. Now I will mark the
point in the interior of angle ACB, which is the intersection of our circle
center at X` and the parallel line passing through X`. LetŐs call this point m.

Now, I want to find a point,
letŐs call it z, on segment AB such that X`m = mz. To do this we construct a
circle about point m with radius equal to the length of X`m. Then I will
construct the intersection of the circle and segment AB.

Now I have three point X`, m
and z and two segments X`m and mz which are equal in length. Thus, I can
construct a rhombus by finding the fourth point, letŐs call it Y`, such that
X`m=mz=zY`=Y`Z`. To do this, I can construct the line passing through point XŐ
that is parallel to mz. The intersection of this parallel line and our circle
about point X`, which is the interior of angle ACB, is our point Y`. Now I have
a rhombus.

Using my rhombus, I can
determine the location of point Y. First I construct the ray AY`. The
intersection of ray AY` and segment CB is point Y. This is the projection of Y`
onto segment CB.

I now need to find X. To find
X, I need to construct a circle about point A with a radius equal to the length
of segment YB. The point where the circle intersects segment AC is point X.

Finally I have point A,B,X
and Y such that AX=XY=YB. The following illustration show circle about points X
and Y with radius equal to the length of XY.