 Assignment 5

GSP Scripts

by

A. Kursat ERBAS

1. Centroid: The centroid of a triangle is the intersection of the three medians of the triangle. A median is the line that passes through the mid-point of one side of the triangle and the opposite vertex. Centroid is generally denoted by G.Click here for a GSP script for Centroid (G) of a triangle ABC.

2. Orthocenter:The orthocenter of a triangle is the intersection of the three altitudes of the triangle. An altitude is the line that passes through a vertex and is perpendicular to the opposite side of the triangle. Orthocenter is generally denoted by H.Click here for a GSP script for Ortocenter (H) of a triangle ABC.

3. Circumcenter:The circumcenter of a triangle is the intersection of the three perpendicular bisectors of the triangle. A perpendicular bisectors is the line that passes through the mid-point of one side of the triangle and is perpendicular to that side. Circumcenter is generally denoted by C. Click here for a GSP script for Circumcenter (C) of a triangle ABC.

4. Circumcircle: A Triangle's circumscribed circle. Its center O is called the Circumcenter, and its Radius R the Circumradius. Click here for a GSP script for Circumcircle of a triangle ABC.

5. Incenter: The Incenter of a triangle is the intersection of the three angle bisectors of the triangle. A angle bisectors is the line that passes through a vertex an divides that interior angle of the triangle in half.Click here for a GSP script for Incenter of a triangle ABC.

6. Incircle: The Inscribed Circle of a Triangle . The center I of the incircle is called the Incenter and the Radius r the Inradius.Click here for a GSP script for Incircle of a triangle ABC.

7. Medial triangle:The medial triangle is the one whose vertices are the midpoints of the sides of a given triangle. Click here for a GSP script for the medial triangle of a triangle ABC.

7a. Orthocenter, Mid-segment triangle: The midsegment triangle is the one whose vertices are the midpoints of the segments AH, BH, and CH (where H is the orthecenter of the triangle ABC). Click here for a GSP script for the medial triangle of a triangle ABC.

8. Orthic triangle: The orthic triangle is constructed by connecting the feet of the altitudes of a triangle ABC. Click here for a GSP script for the medial triangle of a triangle ABC.

9. Pedal triangle: The pedal triangle is created by selecting an arbitrary triangle with an arbitrary point (P) which could be either inside or outside the triangle.Then the intersection of the three perpendicular lines from P to the sides of the triangle creates the pedal triangle. Click here for a GSP script for a pedal triangle of a triangle ABC.

10. Nine Point Circle: The nine point circle is the circle that ,in a triangle ABC, the midpoints of AB, BC, and AC; the points at the feet of the altitudes;and the midpoints of the segments connecting the vertices of triangle ABC to the orthocenter lye on. Click here for a GSP script for the nine-point circle.

11. Center of Nine point circle:The center of the nine point circle lies on Euler's Line midway between the circumcenter and the orthocenter. Click here for a GSP script for the center (N) of a nine-point circle.

12. Trisecting a line segment:Click here for a GSP script for trisecting a given line segment AB.

13. Equilateral triangle, given a side: Click here for a GSP script for the construction of the equilateral triangle, given a side AB.

14. Square, given a side: Click here for a GSP script for the construction of the square, given a side AB.

15. Isosceles triangle, given base and altitude: Click here for a GSP script for the construction of the equilateral triangle, given the base AB and altitude.

16. Triangle Centers (H, G, C, and I): Click here for a GSP script for constructing all four centers (H, G, C and I) of a triangle ABC.

17. Triangle Centers with Euler Line: In any triangle, three remarkable points - circumcenter, centroid, and orthocenter - are colinear, that is, lie on the same line, the Euler line. Centroid is always located between the circumcenter and the orthocenter twice as close to the former as to the latter.Click here for a GSP script for Triangle Centers (H, G, C, and I) with Euler Line.

18. Locus of vertex of a fixed angle that subtends a fixed segment: Locus of vertex of a fixed angle that subtends a fixed segment AB is a circle with radious AB. Click here for a GSP script for the Locus of vertex of a fixed angle that subtends a fixed segment AB.

19. Divide a segment AB into two parts that form a golden ratio: The Golden Ratio () is an irrational number with several curious properties. It can be defined as that number which is equal to its own reciprocal plus one: = 1/ + 1. Multiplying both sides of this same equation by the Golden Ratio we derive the interesting property that the square of the Golden Ratio is equal to the simple number itself plus one: : ^2 = + 1. This gives us either 1.618 033 989 or -0.618 033 989. The first number is usually regarded as the Golden Ratio itself, the second as the negative of its reciprocal. Click here for a GSP script for finding a point R on a segment AB s.t. AR/SR= .

20. Pentagon, given a radius: A pentagon is a five sided regular polygon. Click here for a GSP script for constructing a pentogon with given the radious of its circumcircle. 21. Pentagon, given a side: Click here for a GSP script for constructing a pentogon with given a side.

22. Hexagon, given a side: A hexagon is an eight sided regular polygon. Click here for a GSP script for constructing a hexagon with given a side.

23. Octagon, given a side: Click here for a GSP script for constructing an octagon with given a side. 24. Rhombus, given a side: A rhombus is a parallelogram with all four sides equal. Click here for a GSP script for constructing a rhombus with given a side. 25. Inscribed circle of a rhombus rhombus, given a side: Inscribed circle is the circle tangent to all four sides of a rhombus. Click here for a GSP script for inscribing rhombus. 26. Square, given the diagonal:Click here for a GSP script for constructing a square with given its diagonal. 27. Fermat point: Fermat Point inside a triangle is the point s.t. the sum of the distances from this point to three vertices is the minimum.Click here for a GSP script for constructinf the Fermat Point in a triangle. 28. Inscribing an equilateral triangle: Inscribing equilateral triangle means placing an quilateral triangle inside a circle. Click here for a GSP script for inscribing an equailateral triangle in a circle. 28. Hexagon, given radious: Click here for a GSP script for constructing a hexagon with given radious of its circumcircle. 29. Similar Triangle: Click here for a GSP script for constructing a similar triangle XYZ with given a side XY to a given triangle ABC. 30. Octagon, given a radious: Click here for a GSP script for inscribing a regular octagon in a circle given its radious. 31. A square with area twice the area of a square with one side AB: Click here for a GSP script for constructing a square with area twice the area of a square with one side AB. 32. A square with area three times the area of a square with one side AB: Click here for a GSP script for constructing a square with area three times the area of a square with one side AB. 33. Logarithmic Spiral (Equiangular Spiral, Golden Spiral, Spira Mirabilis):

Equiangular spiral is a curve that cuts all radii vectors at a constant angle.

Explanation:
1.Let there be a spiral (that is, any curve r==f[theta] where f is a monotonic function)
2.From any point P on the spiral, draw a line toward the center of the spiral. (this line is called the radial vector)
3.If the angle formed by the radial vector and the tangent for any point P is constant, the curve is an equiangular spiral.

Note: After using the script

Drag point D to change the constant angle.
Drag point M to increase/decrease range of curves ploted.

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