## GSP Script Tools

### By Annie Sun

Below are short cut tools for completing a construction in GSP.  The tool, its definition, and construction details are available in the table below. Additional tools are available at the end.

 GSP Script Tool Definition Construction Details Centroid The intersection point, G, of the medians of a triangle Begin by constructing any triangle, and then find the median (midpoint) of each side.  Next, connect the medians of each side of the triangle with the opposite vertices of the triangle.  The intersection point of the medians in called the centroid and typically denoted with G. Orthocenter The intersection point, H, of the altitudes of a triangle. Begin by constructing any triangle.  Then, create the perpendicular lines from each vertex of the triangle to each side of the triangle (altitude).  The intersection point of the altitudes is called the orthocenter and typically denoted with H. Circumcenter The point equidistant from the vertices of a triangle, C. Begin by constructing any triangle.  Then, create the perpendicular bisector of each side of the triangle.  The intersection point of the perpendicular bisectors is called the circumcenter and denoted by C. Circumcircle The circle that inscribes the triangle and uses the circumcenter of the triangle as its center. Begin with the circumcenter construction above of any triangle.  Then, using the circumcenter as the center of the circle, construct the circumcircle through the vertices of the triangle. Incenter The point equidistant from the sides of a triangle, I. Begin by constructing any triangle.  Then create the angle bisector of each of the three angles of the triangle.  The point at which the angle bisectors intersect is the incenter and denoted with I. Incircle The circle that is inscribed in the triangle and uses the incenter as the center. Begin with the incenter construction above of any triangle.  Then, from the incenter, construct a perpendicular line segment to any side of the triangle; use the intersection point to construct the incircle through that point with the incenter as the center. Medial Triangle The triangle inside a given triangle that uses the medians of the given triangle as vertices. Begin by constructing any triangle.  Then, determine the midpoints of each side of the triangle.  Draw 3 line segments to connect the midpoints; the median triangle is formed. Mid Segment Triangle The triangle created using the orthocenter of a given triangle and the midpoint of the orthocenter to the vertices of the given triangle. Begin by constructing any triangle and its orthocenter, detailed above.  Then, construct line segments from the orthocenter to the vertices of the original triangle; determine the midpoints of these segments.  Connect the midpoints with line segments, which will then become vertices of the mid segment triangle. Orthic Triangle The triangle inside a given triangle that connects the altitudes of each side. Begin by constructing any triangle.  Then, create the perpendicular lines from each vertex of the triangle to each side of the triangle (altitude).  Connect the points where the perpendicular lines intersect the sides of the triangle; this will give you the orthic triangle. Pedal Triangle The triangle made up of a random point on the plane and perpendicular intersections to another triangle. Begin by constructing any triangle; extend the sides of the triangle to make lines.  Then place a point anywhere in the plane, this will be your pedal point.  From the pedal point, construct perpendicular lines to each of the sides of the triangle (you may need to use the extended lines here).  Connect the intersection points made by the perpendicular lines; the pedal triangle is formed. Nine Point Circle The circle formed using the orthocenter and circumcenter of a triangle. Begin by constructing any triangle and its orthocenter and circumcenter, using the construction details above.  Then construct a segment connecting the orthocenter and circumcenter; find the midpoint of the segment.  Construct a circle using the midpoint of the orthocenter and circumcenter as the center and any point related to the aforementioned constructions as points on the circle; the nine point circle is formed. Trisecting a Line Segment This tool gives a line segment that has been divided into three equal parts. Begin by constructing a line segment AB.  Find the midpoint of the line segment AB and label is C.  Construct the circle BC with point B as the center and C as a point on the circle.  Extend the line segment AB to a line and mark its intersection point on the circle BC as D.  Line segments AC, CB, and BD are now of equal lengths. Equilateral Triangle A triangle with all sides equal to the same length Begin with any line segment AB, then construct a circle with A as the center and B as a point on the circle.  Construct another circle with B as the center and A as a point on the circle.  Choose one of the points where the two circles intersect to be the third vertex of the equilateral triangle; and connect the vertices. Square A quadrilateral with four equal sides. Begin with a line segment AB.  Then, construct a perpendicular line to the line segment AB passing through point A; construct another perpendicular line to the line segment AB passing through the point B.  Next, construct a circle with center A and passing through the point B.  Construct another circle with center B and passing through the point A.  The perpendicular lines will intersect with the circles on opposite sides of the line segment.  Connect one pair of the intersection points with a line segment and you will have a square. Isosceles Triangle A triangle with two sides that are equal in length Begin with a line segment AB.  Then find the midpoint of AB and construct a perpendicular line, through the midpoint.  Create another line, CD, that will be used as the altitude of our isosceles triangle.  Construct a circle with center on the midpoint of AB and using CD as the radius.  The circle and perpendicular line will intersect in two places, choose one for the third vertex of the triangle.  Connect the vertices and you will have an isosceles triangle. Triangle Centers A triangle with the centroid, orthocenter, incenter, and circumcenter already constructed Combine the constructions above for centroid, orthocenter, incenter, and circumcenter. Euler Line A line in which the centroid, orthocenter, and circumcenter’s of a triangle lie on Use the construction of the triangle centers and construct a line through the centroid and orthocenter. Locus The set of all points equidistant from a given line segment Begin by constructing any triangle.  Then choose the vertex which will serve as the locus that will subtend one of the sides.  Create an arc with the chosen vertex as the middle point. Golden Ratio A line segment with the proportions of the golden ratio Begin with a line segment AB, then find the midpoint of AB.  Construct a perpendicular line to AB through the point A.  Then, construct a circle with midpoint A passing through the midpoint.  Make the point of intersection of the perpendicular line and the circle point C.  Construct a circle with center at point C, passing through point A.  Create a line segment connecting point C with point B; this line segment will intersect with the circle centered at C and label the intersection point D.  Then, construct a circle with center B, passing through the point D; this circle will intersect the line segment AB with the golden ratio. Pentagon, given a radius A pentagon constructed by a given radius Pentagon, given a side A pentagon constructed by a given side Hexagon A six sided polygon with sides of equal length constructed from a given side Octagon An eight sided polygon with sides of equal length constructed from a given side

External Tangent Lines for Two Disjoint Circles

Internal Tangent Lines for Two Disjoint Circles

Tangent Circles (1)

Tangent Circles (2)

Center of a Nine Point Circle