**Other Constructions: GSP Tools**

by

**Sarah Major**

Trisecting a Line Segment

Begin with an ordinary line segment. Construct another segment extending from the first endpoint of the segment. Make sure the new segment is relatively long. Find the midpoint of this new segment. Then find the midpoints of the two segments resulting from splitting this segement in half. Construct perpendicular lines from the second segment to the first segment, and use the points resulting from the three midpoints. The intersections of these perpendicular lines with the original segment have resulted in a trisection.

**Segment Split Into the Golden Ratio**

Draw a segment. Find its midpoint to create a length that is half of the segment. Draw a circle with a center at B and a radius that is this half length. Draw a perpendicular through B. Where this line intersects the circle, draw a segment from this point to A. Construct the same sized circle with a center at this intersection point. Draw a circle with a center at A passes through the point where the previous circle intersected the hypotenuse of the triangle. Where this circle intersects the segment is the point that splits the segment into the golden ratio.

**Locus of a Vertex of a Fixed Angle that Subtends a Fixed Segment**

First, construct an angle ABC and a segment DE. Find the midpoint of DE, which is F. Draw lines perpendicular to DE through both D and F. Construct a circle with center D and a radius of AB from the angle. Where the line DE intersects the circle is to be labeled G. Then, draw a segment AC, officially making the original angle a triangle. Construct a circle with center G and radius AC. Label the intersection of this circle with the first circle as H (I did the right intersection, do the intersection that will make EDH an acute angle). Construct a ray starting at D and going through H. Where this ray intersects the perpendicular line DG is to be labeled I. Draw the segment ID and use it to construct a circle with center I and radius ID. Pick a random point on this circle (making sure it is not any of the intersections) and label this J. Construct the segment DJ and the segment JE. Construct an arc through EJD. This is the locus.

Parabola

Construct a line. This will be the directrix. Construct a point somewhere above or below the directrix. This will be the focus. Construct a line perpendicular to the directrix. Where this intersects will be the animated point. Also, construct a line from this point to the focus. Find the midpoint. Draw a line perpendicular from the line from the focus to the directrix passing through this midpoint. Where this line intersects the line perpendicular to the directrix will be traced. The trace is the parabola.