Geometry Explorer Lab F2. Edited F2 after class. >>>Note, edited portions are marked by a >>> so you don't have >>>to check every line. Choose group members (3-4). Exchange email/phone coordinates. Choose name of group: Thales, Pythagoras, Euclid, Archimedes, Pappus, Klein, Poincare, Hilbert, Birkhoff. Elect Presenter, Secretary, Demonstrator. Record your experiments and prepare to write up a report. A. Euclidean Geometry In this exercise we explore Euclid's Five Postulates in all the models GEX 2.0 gives us. See p403 of Hvidsten E1: To draw a straight line from any point to any point. E2: To produce a finite straight line continuously in a straight line. E3: To describe a circle with any center and distance. E4: That all right angles are equal to one another. E5: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side side on which are the angles less than two right angles. Our translation: E1: Two points determine a (unique) line (segment) between them. E2: This segment lies on an infinite line. E3: Draw a circle with given a radius (center and point on circle) >>> E4: ???????? E4: (Perpendiculars) From a point outside a line drop the perpendicular. E5: (Playfair's Postulate). Given a line and a point not on the line there is a unique line through the point parallel to the given line. (Lines with no common points are called parallel.) >>> I've decided to substitute Euclid's Proposition 12 for E4. Thus our >>> working versions of the postulates all say you can construct something. Models: Euclidean: Plane, Central Ortho, Central Stereo, Quad Model Suggestion: Start with Central Stereo, and switch to Quad. You can continue to construct in the top left view, and see what it looks like in all three. Hyperbolic: Poincare Disk, Klein Disk, Upper Half Plane, Quad Elliptic: (This part of GEX 2.0 is not complete yet. For now there is only one view Vocabulary: Wiggle: Choose an object (e.g. a point) and move it around. Produce: Choose endpoints of a segement > Transfrom Mark > Vector > 2 Pts Choose segment > press Translate button. B. Exterior Angle Theorem Make this construction: 1. Choose an arbitrary triangle ABC with exterior angle DBC by 2. producing base AB one length beyond B to D.. 3. Construct the median (midpoint of the opposite side) from A to A' on BC >>> and produce is one length into the exterior angle. 4. Complete connect A"B to form exterior triangle A"A'B and prove that it is congruent to triangle AA'C by SAS. (Watch the order of vertices!) 5. Conlude the angle ACB is congruent to angle A"BC. 6. Observe that the opposite interior angle at C has a congruent image inside the exterior angle DBC. So it is smaller. 7. Repeat the construction by extending median CC', where C' is the midpoint of AB. 8. What is your conclusion in Eucliean, Hyperbolic and Elliptic geometry? A report on labF2 is due F3.