Advanced Euclidean Geometry

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$University of Colorado Denver Math 1070 Topics Updated: Fall 2009 ...$

$Math 3000 Running Glossary$

9-point Circle Thm 4.17: The centroid of a triangle trisects the segment joining the circumcenter and the orthocenter. [Euler line] A H G C A' O B Draw centroid G on median AA', 2/3 of the way from A. ΔHAG ~ ΔOA'G because AH = 2OA' AG = 2GA' and ∠HAG ≅ ∠OA'G thus ∠HGA ≅ ∠OGA', and so H,G, and O are collinear with HG = 2 GO.
9-point Circle Thm 4.18: The center of the nine-point circle bisects the segment of the Euler line joining the orthocenter and the circumcenter of a triangle. B H A J A' O Recall from the last proof that AH = 2OA'. Since J is the midpoint of AH, we have JH = OA'. Thus, JOA'H is a parallelogram. But JA' is a diameter of the 9-point circle, and also a diagonal of this parallelogram (HO being the other diagonal). Since the diagonals of a parallelogram bisect each other we have the conclusion. C
Page 1 and 2: Advanced Euclidean Geometry
Page 3 and 4: Circumcenter Equally far from the v
Page 5 and 6: Examples: Circumcenter (O)
Page 7 and 8: Orthocenter (H) Thm 4.2: The altitu
Page 9 and 10: Equally far from the sides? A B C I
Page 11 and 12: Examples: Incenter (I)
Page 13 and 14: Internal Bisectors Thm 4.5: The int
Page 15 and 16: A Circle Theorem Thm 4.9: The produ
Page 17 and 18: Directed Segments The next few theo
Page 19 and 20: I B' II A D E A' III B C Menelaus'
Page 21 and 22: Ceva's Theorem Thm 4.13 : Ceva's Th
Page 23 and 24: Ceva's Theorem Note that the text d
Page 25 and 26: Gergonne Point Thm 4.14: The lines
Page 27 and 28: Miquel Point Thm 4.19 : If three po
Page 29 and 30: Feuerbach's Circle Thm 4.16 : The m
Page 31 and 32: The 9-Point Circle Worksheet 2. Fou
Page 33 and 34: 9-Point Circle Worksheet A trapezoi
Page 35 and 36: 9-Points Circle Worksheet 4. In a r
Page 37 and 38: 9-Points Circle Worksheet Since A'D
Page 39: 9-point Circle Thm 4.17: The centro
Page 43 and 44: Symmedian In a triangle, lines thro
Page 45 and 46: c b B a C Law of Sines A a sin A =
Page 47 and 48: Theorem Theorem: The isogonal conju
Page 49 and 50: Morley's Theorem The incenter I of
Page 51 and 52: Draw the sides of ΔABC and note th

Advanced Euclidean Geometry

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