By Alfred S. Posamentier
Advanced Euclidean Geometry provides an intensive evaluation of the necessities of high university geometry after which expands these recommendations to complicated Euclidean geometry, to provide academics extra self assurance in guiding pupil explorations and questions.
The textual content includes hundreds of thousands of illustrations created within the Geometer's Sketchpad Dynamic Geometry® software program. it really is packaged with a CD-ROM containing over a hundred interactive sketches utilizing Sketchpad™ (assumes that the person has entry to the program).
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Additional info for Advanced Euclidean Geometry
Prove that points P, Q, C, and P are concyclic. As you proceed through the rest of this book, you may want to work with additional exercises. For this purpose you might use Challenging Problems in Geometry by A. S. Posamentier and C. T. Salkind (New York: Dover, 1996). , three or more lines containing a common point) usually gets a light treatment in an elementary geometry course because of higher priorities. Acquiring a truly good facility with the concept would require that more theorems be explored than time permits in the first geometry course.
This last example of duality demonstrates that related words also need to be changed when forming the dual of a statement. Specifically, note that collinear and concurrent are dual words, as are triangle and trilateral. Recall Cevas theorem (see Figure 3-1): 43 44 ADVANCED EUCLIDEAN GEOMETRY The three lines containing the vertices A, B, and C of AABC and intersecting the opposite sides at points I, M, and N, respectively, are concurrent if and ^ , A N BL CM For the most part, the dual of a postulate is also a postulate, and the dual of a definition is itself a definition.
By addition, mAAOP = mABOP. The supplements of these angles are also equal in measure: mAAON = mABON. But because AAOQ = ABOQ (SSS), mAAOQ = mABOQ. Because the angle bisector is unique, ON and OQ must coincide and the perpendiculars to these must also be parallel. Hence AB 11 CD. • Repeat the “proof” for O outside the quadrilateral. Then repeat the proof for O on DC. 22 ADVANCED EUCLIDEAN GEOMETRY 2. Discover the fallacy in the following “proof”: 45° = 60°. “P r o o f ” Construct equilateral triangle ABC.
Advanced Euclidean Geometry by Alfred S. Posamentier