Some Useful and Interesting GSP Script Tools

(I worked on this assignment through the whole semester of fall 2009, finally got it done 11th on Nov.16,2009)

Chen Tian

**Some script tools for a problem (made up by myself) about shooting balls on a pool table**:

First we construct a Rectangle as shown in the GSP document.

Then we construct the midpoint of each of the two longer sides. Also we plot a white point inside the rectangle.

Now we get an abstract Pool Table.

Let's set some balls on the pool table in the way shown in the picture.

What is the route of the object ball after being hit by the cue ball (no side spinning) on the center of its front surface (ignoring the friction, air resistance, etc.) assuming the object ball will hit one of the cushions of the pool table (or else the route will be a ray)? Use this construction we won't need protractor or reflection. (Why?)

What if the object ball hits more than one cushion? We can use the image point of the object all, and the hitting point on the cushion or we can pick up more any point on the route (after the ball hit the first cushion) inside the pool table, and then use the same script tool above the decide the continous route of the ball. (Why? Students please think about the reason.)

Now fun time! We can play around to see which object ball will mathematically be potted into the pockect or never.

Here is an example of the routes of some color balls. Please use the "hide/show" bars.

By the way, the real billiards or snooker will be much more compilcated than this. But the idea to play pool games would be based on the above mathematical thinking.

Oh, here is a complete document I made which was used for a mathematics problem presentation.

**Now some other useful script tools**:

In two ways trisect a given segment.

Construct the centroid of a triangle, which is the intersection of the three medians of the triangle.

Construct the medial triangle of a given triangle.

Construct the triangle of medians of a given triangle.

Construct the circumcenter of a triangle, which is the intersection of the three perpendicular bisectors of the three sides of the given triangle.

Construct the circumcircle of a triangle.

Construct the incenter and the incircle of a given triangle.

Construct the orthocenter and the orthic triangle of a given triangle.

Construct the nine-point circle of a given triangle.

Construct the pedal trianlge of a given triangle. (Definition: Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars, through P, to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.)

Locus of vertex of a given angle which fitts in a given circle.

Locus of vertex of a fixed angle that subtends a fixed segment.

Construct the golden ratio segment of a given segment.

Divide a segment AB into two parts that form a golden ratio.

Given a radius, construct a regular pentagon (a pretty fundamental construction using Euclid's original idea, instead of directly using the golden ratio script tool created above).

Given a side, construct a regular pentagon (a quick construction directly using the golden ratio script tool created above).

The constructions of a regular exagon given a side, and a regular octagon given a side would be pretty easier than that of a regular pentagon, since it only involves the simple construction of an equilateral triangle or a square. (Leave these to students who I assume should know how to do them.)

Construct a circle tangent to the other two. (Please pay attention to the locii of the center C of the circle when moving point D along the circumference of circle A and changing the relative positions of circle A and circle B. (To move point D, we can use the function "animation objects" under "display" in the tool bar of Sketchpad. To trace point C, we can use the function "trace" under "display" in the tool bar of Sketchpad.)

Draw a third circle decided by the first two circles with their tangents at the same intersection point.

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