LAB F2. KSEG is a free-ware Euclidean plane geometry drawing tool unlike the Geometry Explorer and the Geometer's Sketchpad, to name two licensed packages. In the mac-labs, 24IH, 239AH, 102AH you will find KSEG on the "dock" at the bottom of the screen. Since KSEG is free, you can also download a copy to own PC. Here is how you download and install it on your PC: Google > kseg > download (for windows) > save to disk > unzip (extract all) In these notes we will use a shorthand to simplify the recipes. We use only the key word to identify the name, button, menu, or whatever you can click, drag or choose. The > means the next step. This will place the KSEG folder on your desktop, which is actually located at C:\docume~1\\Desktop\ (here "docume~1" is the folder usually called "Documents and Settings".) You do have to extract the zipped up package, don't try to run it from the zipped package. These notes are based on kseg-0.401. You may have a newer version. Feel free to explore and google for more information. Here we treat only the absolutely least necessary step for you to make the constructions. Lab Exercise F2 1. Centroid Theorem: [Note, 4sep09 you have already done this exercise in class. So if you still have it on your desktop, just show me that construction for credit.] Choose three points (right mouse) > select all three (hold shift key down as you choose more than one thing at a time) > click on the segment tool > choose segment > click on midpoint tool > contruct two of the medians > select both and click on intersection tool > construct the third median. The philosophy behind KSEG is good geometry. You need two points, not one or three, to make segement. Only segments have midpoints. The selected object is outlined in red. To shake stuff off your mouse-paw, click in an empty space. (This is an annoying feature of KSEG, but hey, it's free!) You can move a constructor (one vertex of a triangle, for example) to change the picture. We call this "wiggling" the figure. When a feature persists (e.g. the medians are concurrent), you have a "computer proof" of a theorem. 2. Octahedron inscribed in a quadrilateral. Choose four points > make a quadrilateral > construct the diagonals > wiggle the figure until you see it as a projection of a 3-D tetrahedron > bisect all 6 sides > connect four of them up in a parallelogram > do it again (you may have to wiggle to get a good figure) > do it the third way and wiggle the figure until you can see the octahedron. 3. Save your work to the desktop > transfer it to your netfiles directory. This way you can continue work on any PC you happen to sit in front of. 4. Memorize the download procedure so you can install KSEG if necessary. Lab Exercise F2bis [You will need this construction in the future. Today I can help you with it. But you do not need to finish it today. Save your work.] Here you will first construct Euler's line of a triangle, and then Feuerbach's circle. Don't get discouraged, this takes some time to get used to. Note that I will use labels that may be different from the labels KSEG automatically assigns vertices. You can reassign the label, but that could be done at the very end, since it is tedious. Better use paper and pencil to keep track of what you're doing. 1. Construct a triangle ABC. Locate its centroid G (I don't have to tell you again how to do this, you've already done this construction.) You'll need only two medians, because you already know that the third median goes through the same point. But now you need to construct the point where the two medians cross. Do this by selecting both lines > click on the intersection tool. You can hide the medians, but keep the centroid. Next find the circumcenter D. (You'll need to find where two perbises cross. Then hide the perbises from view.) To extend a given segment, DG, you could use a ruler, but there is a more geometrical way of doing this. Select D and G in that order > click on the vector tool > choose G and translate it by the vector DG>. This doubles the segment, but we want to go twice as far. So, translate the endpoint once more to get to a new point H. In vector notation, H = D + 2 (G-D). Now draw one median, say it is AA' (but it could have been any of the other two). Draw the line segments DA' and AH, and find two triangles. Show, by measuring the angles, that the two triangles are similar. Be careful, the angle measure of AGH is not the same as the angle measure HGA, one will be the negative of the other. But KSEG writes -30 as 360-30=330. The similarity shows that AH is parallel to DA', therefor the line (AH) is the altitude from A. Wiggle this picture and see what happens when ABC becomes obtuse! 2. Next, construct the Feuerbach's Nine Point circle. You will need the segment HD, but you won't need the centroid G anymore. Construct the midpoint N of HD (i.e. N=(H+D)/2). To build a circle centered at N with the point A' on it, for example, first choose N > select it as center > choose A' > click on circle tool. Identifiy all nine points on this circle. For this you'll need to draw all three altitudes and bisect AH, BH and CH. This exercise should remind you of high school biology when you dissected a frog. 3. Be sure you save your work. Wiggle the figures until they look nice. You can print a .pdf or a .jpg on paper and give it to me for credit for this lab. Be sure you do a figure for an obtuse as well as acute triangles. If you are handy, you can get several images on a single sheet of paper, or maybe one sheet for each exercise.