Congruent Segments

By: Ginger Rhodes

Problem: Given three points A, B, and C. Draw a
line intersecting AC at point X and BX at point Y such that AX = XY = YB.

Let’s
begin with three points A, B, C.

Next,
we will construct a smaller version of the congruent segments. To begin we will
construct a point D on the segment AC. Then construct a line parallel to
segment CB through the point D. Now, how can we create a segment congruent to
segment AD through the point D on the parallel line?

Now,
we would like to construct a segment from point E to segment AB and congruent
to segments AD and DE. So let’s construct a circle with center E and
radius length DE.

Next,
we will construct a rhombus with sides DE and EF.

The
segments AD, DG, and GF are all congruent. Since our figure is getting
complicated we will hide objects that we don’t need anymore, such as the
circles and the rhombus. Then we will construct a line through the point A and
G. Label the point of intersection of the new line and segment CB as Y. Now,
construct a line parallel to segment DG through the point Y. What triangles are
similar?

Since
parallel lines we know corresponding angles are congruent. So <ADG@<AXY, <AGD@<AYX, <AGF@<AYB, and <AFG@<ABY. Therefore, DADG~DAXY and DAGF~DAYB because AA~.

Since
there are similar triangles we know corresponding sides are proportional, so

By
noticing both triangles share sides AG and AY we know

Since
AD = DG = GF, we can conclude AX = AY = YB.