Christa Marie Nathe
Faith In Construction
The objective in this investigation is to locate a point x on line AB such that when connected with point y on line BC there will be three equal line segments. The following construction shows the relationship described above, where we desire for the line segments of Ax, xy, and yC to be equal.
The subsequent steps and related figures will show how to find points x and y such that there position will yield three equal line segments.
Start by creating an angle and labeling it ABC
Now create a line from A to C, to make the angle into a triangle and plot a random point P on line BC.
The line AC was created to give us a familiar shape to work with, of which it might help to influence our ability to find the desired points x and y. Point P is not a fixed position, it is there in order to give us something to manipulate in our figure.
Constructing a circle with the point P as the center with radius PA we can develop more geometric relationships within the figure. In addition, we can use draw a line parallel to BC through point P which gives us yet another geometric association.
The previous figure tell us that PQ=PA.
By creating another circle with its center at Q, with the same radius of PQ, we can take full advantage of the length of the radius to help determine where point x ought to be placed. Currently, we cannot manipulate the radius or the circle to do anything useful.
This ensemble is rich with associations as we can see. The two circles with equal radii allow us to form a rhombus. By connecting PQR to form a triangle and reflecting it over the line PR we obtain a rhombus. Lucky for us, a rhombus has two characteristics that work in our favor; its sides are equal in length (in our case, the radius of the circles) and its opposite sides are parallel. As you remember, the light blue dashed line was created parallel to BC. If segment PQ is parallel to BC, then that means that SR is parallel to both BC and PQ that aids in our quest to find the three equal line segments. We have thus far established some helpful equivalent relations.
By dragging point P, remember it is not fixed, down the line AB segment, we are increasing the size of the circle therefore its radius, which directly affects the sides of the rhombus. We can position P such that the radius PA=PS=SC of the rhombus.