6. 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
The most important thing to realize about this problem is that you are going to have to use the idea of similar triangles. First of all, let's start by drawing line segments AC and BC.
Now, we need to put in a random point D on AC. (Notice that this is not the X point that we will need for the answer, but it is a point that we will use for similar triangles.) Let's also create a circle with D as the center and with A lying on the circle.
Next, we need to create a line passing through D that is parallel to BC. Let's label the point that intersects the line and circle as E. Also, let's go ahead and construct AB.
Notice that AD and DE both have the same length since they are radii of the same circle. Now, let's construct a circle with center E and radius DE. Let's label the intersection of this circle and AB as F, and let's construct segments, AD, DE, and EF.
Notice that EF has the same length as the other two segments. Now let's create a rhombus with DE and EF being two of the sides. Let's also get rid of some of the "junk" that we don't need.
Can you see that AD = DG = GF? If only we had been given line HF instead of line BC! How can we use this picture to find the missing X and Y points? The answer is . . . . similar triangles!!!
By constructing parallel lines where applicable, we have constructed two similar triangles, ADG and AXY. So since AD = DG, we know that AX = XY. Also, notice that triangle AGF is similar to triangle AYB. So GF is similar to YB. And since AD = DG = GF, we know that AX = XY = YB!! Click here to see for yourself!!