Advanced Euclidean Geometry

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

$Math 3000 Running Glossary$

9-Points Circle Worksheet We are now ready to discuss the Feuerbach circle. For an arbitrary triangle, the 3 midpoints of the sides, the 3 feet of the altitudes and the 3 points which are the midpoints of the segments joining the orthocenter to the vertices of the triangle all lie on a circle, called the nine-points circle. There is a circle passing through the 3 midpoints of the sides of the triangle, A', B' and C'. We shall show that the other 6 points are on this circle also. Let D be the foot of the altitude from A. Consider the quadrilateral A'DB'C'. We claim that this is an isoceles trapezoid. r) Why is A'D parallel to B'C' ? s) Why is DB' equal in length to 1/2 AC? t) Why is A'C' equal in length to 1/2 AC?
9-Points Circle Worksheet Since A'DB'C' is an isosceles trapezoid, D must be on the same circle as A', B' and C'. The other altitude feet (E and F) are dealt with similarly. u) Determine the isosceles trapezoid that contains E and the one that contains F. Now consider J, the midpoint of the segment joining the orthocenter H to the vertex A. Draw the circle that has A'J as its diameter. v) Why is A'B' parallel to the side AB? w) Why is JB' parallel to the altitude CF? x) Show that A'B' is perpendicular to JB'.
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: 9-Points Circle Worksheet 4. In a r
Page 39 and 40: 9-point Circle Thm 4.17: The centro
Page 41 and 42: 9-point Circle Thm 4.18: The center
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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