Download PDF by Alfred S. Posamentier: Advanced Euclidean Geometry

Algebraic Geometry

By Alfred S. Posamentier

Advanced Euclidean Geometry provides an intensive evaluate of the necessities of high institution geometry  after which expands these ideas to complicated Euclidean geometry, to offer academics extra self assurance in guiding scholar explorations and questions.

The textual content comprises 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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DE and GE intersect at point P. ^ in g BC as a side, construct parallelogram BCJK so that BiC 11PA and BK = PA. From this configuration. d . 300) proposed an extension of the Pythagorean theorem. He proved that the sum of the area of parallelogram ABGF and the area of parallelogram ACDE is equal to the area of parallelogram BCJK. Prove this relationship. ) FIGURE 1-34 Chapter 1 ELEMENTARY EUCLIDEAN GEOMETRY REVISITED 23 4. GIVEN: BE and AD are altitudes (intersecting at H) of AABC, while F, G, and K are midpoints of AH, ABy and RC, respectively (see Figure 1-35).

In any case, we would at least have a statement that would be a good candidate to be a theorem. A valid proof would be needed to establish the statement as a theorem. This is precisely what we will now investigate. With our knowledge of dual­ ity, we will form the dual statement of Ceva’s theorem. Actually, it was the rediscovery of Menelaus of Alexandria's famous but forgotten theorem,^ which we will discuss in the next section, that led Giovanni Ceva in the first book of his De lineis rectis se invicem secantibus statica constructio (Milan, 1678) to pro­ duce his theorem by the principle of duality.

4. In AABC (Figure 2-15),_^,_BM, and CN are concurrent at point P. Points R, 5, and T are chosen on EC, AC, and AE, respectively, so that NR || AC, LS IIAE, and MT || EC. Prove that AR, E5, and CT are concurrent (at point Q). Chapter 2 CONCURRENCY of LINES in a TRIANGLE 39 5. In AABC (Figure 2-15), AL, BM, and CN are concurrent at point P. Points Uy Vy and W are chosen on AJ5, AC, and BCy respectively, so that LU\\ AC, N V II BCy and MW' || AR. Prove that AW, RV, and CU are concurrent (at point K).

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