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Similarity, Ratios,
Proportions - Table of
Content
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86. |
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Euclid's
Elements Book I, 23 Definitions. One-page visual illustration.
Euclid's
Elements Book. Index |
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85. |
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Euclid's Elements Book VI, Proposition 3: Angle Bisector Theorem
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84. |
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Right
Triangles, Formulas and Facts
Pythagorean theorem, Similarity, Geometric mean,
hypotenuse, legs, semiperimeter, altitude, median,
similar shapes areas, centers, orthocenter, incenter,
centroid, circumcenter, radius, inradius, circumradius,
exradius, circles, incircle, circumcircle, excircle. |
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83. |
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The significance of the Pythagorean theorem by Jacob Bronowski.
Pythagorean Theorem, 47th Proposition of Euclid's
Book I. |
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82. |
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Mascheroni Construction with
compass alone. Midpoint of a segment.
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81. |
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Stewart Theorem Triangle and a cevian.
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80. |
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Heron's Formula.
Key facts and
a purely geometric
step-by-step proof.
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79. |
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Heron's Formula
with medians |
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78. |
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Gergonne Point Theorem.
Concurrency.
Interactive proof with animation.
Key concept: Ceva's
Theorem.
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77. |
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Ceva's Theorem.
Concurrency.
Interactive proof with animation. Key concept:
Menelaus Theorem.
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76. |
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Puzzle of
the Ceva's Theorem: 48 classic piece. |
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75. |
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Menelaus' Theorem.
Interactive proof with animation and key concepts. |
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74. |
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Gergonne's Theorem: Triangle and Concurrent Cevians. |
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73. |
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Van Aubel's Theorem II: Triangle and Cevians.
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72. |
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Proposed Problem
64: Triangle, Incircle, Transversal. |
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71. |
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Triangle with the bisectors of the
exterior angles. Collinearity. Key concept:
Menelaus Theorem.
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70. |
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Ptolemy's Theorem.
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69. |
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Cyclic
Quadrilateral: Ratio of the Diagonals
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68. |
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Casey's Theorem. Generalized Ptolemy's Theorem.
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67. |
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Pascal's Mystic Hexagram Theorem Proof |
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66. |
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Eyeball Theorem:
Animated Angle to Geometry Study. |
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65. |
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Butterfly Theorem
Proof with animation. |
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64. |
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Nine-Point Center,
Nine-Point Circle, Euler Line.
Interactive illustration.
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63. |
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Monge &
d'Alembert Three Circles Theorem II with Dynamic Geometry. You can alter the geometric construction dynamically in order to test
and prove (or disproved) conjectures and gain mathematical insight that
is less readily available with static drawings by hand. Requires Java
Plug-in 1.3 or higher. Please be patient while
the applet loads on your computer. |
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62. |
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Monge &
d'Alembert Three Circles Theorem I with Dynamic Geometry. You can alter the geometric construction dynamically in order to test
and prove (or disproved) conjectures and gain mathematical insight that
is less readily available with static drawings by hand. Requires Java
Plug-in 1.3 or higher. Please be patient while
the applet loads on your computer. |
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61. |
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Blanchet Theorem
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60. |
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Lemoine Theorem |
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59. |
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Nagel Point
Theorem. Proof.
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58. |
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Pentagons
& Pentagrams. I found, exploring with dynamic geometry software, new
facts about pentagons and pentagrams. Key concept: Menelaus Theorem.
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57. |
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Interactive Hinged tessellation. Square tessellation - Dynamic Geometry.
Please be patient while the applet loads on your computer. This Java applet requires Java 1.3 or higher. |
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56. |
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Proposed Problem 62: Square Diagonal, Inscribed Circle. |
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55. |
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Proposed Problem
40.
Triangle, Incenter, Excenter, Angles 80, 40, Distances. |
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54. |
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Proposed Problem
40. Geometry Help.
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53. |
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Sangaku Proposed
Problem 41. "Mickey Mouse."
Three circles, tangents, radii. |
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52. |
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Proposed Problem 63: Heptagon Regular, Side and Diagonals. |
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51. |
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Morley's Triangle & Center: with interactive animation and
manipulation.
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50. |
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Newton's
Theorem: Newton's Line. Circumscribed quadrilateral, midpoints of
diagonals, center of the circle inscribed.
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49. |
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Proposed Problem
70: Squares inscribed in a triangle, Similarity.
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48. |
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Proposed Problem
69: Square inscribed in a triangle, Similarity.
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47. |
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The Pythagorean Curiosity
Triangles and squares, fifteen conclusions. |
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46. |
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Complete Quadrilateral: Ortholine-Steiner Line.
Step-by-Step construction, Manipulation, and animation |
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45. |
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Newton's
Theorem, Newton-Gauss Line: Complete quadrilateral theorem. Using TracenPoche Dynamic
Geometry Software, Online
Step-by-Step construction, manipulation, and animation. |
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44. |
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Archimedes' Book
of Lemmas. Exercise your brain. Archimedes wrote the "Book of
Lemmas" more than 2200 years ago. Solve these 15 high school level
problems and lift up your geometry skills.
Proposition 4, Arbelos, |
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43. |
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Archimedes' Book
of Lemmas.
Proposition 5, Arbelos, Archimedean Twins.
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42. |
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Archimedes' Book
of Lemmas.
Proposition 6,
Arbelos,
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41. |
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Archimedes' Book
of Lemmas.
Proposition 7, Similarity of Areas.
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40. |
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Archimedes' Book
of Lemmas.
Proposition 10, Circle, Tangents,
Chord, Parallel
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39. |
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Archimedes' Book
of Lemmas.
Proposition 11, Perpendicular chords
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38. |
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I have used
Geometry Expressions, the world's first Interactive Symbolic
Geometry System, to visualize the Archimedean Twins and check out a
variety of conjectures.
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37. |
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Proposed Problem
79: Triangle.
Similarity, Altitudes, Orthocenter, Incircles, Inradii. |
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36. |
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Proposed Problem
131. Van Aubel Theorem, Concurrent Cevians, Sum of Ratios.
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35. |
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Proposed Problem
130. Triangle, Concurrent Cevians, Sum of Ratios.
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34. |
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Proposed Problem
129. Triangle, Concurrent Cevians, Sum of Ratios.
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33. |
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Proposed Problem
128. Incenter of a Triangle, Angle Bisectors, Sum of Ratios.
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32. |
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Proposed Problem
127. Centroid and Incenter of a Triangle,
Parallel, Proportions.
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31. |
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Proposed Problem
126. Incenter of Triangle, Angle Bisector, Proportions.
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30. |
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Marion Walter's Theorem: Triangle and Hexagon areas: Using TracenPoche Dynamic
Geometry Software |
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29. |
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PProposed Problem
125. Area of Triangle, Star,
Trisection of Sides.
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28. |
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PProposed Problem
124. Area of triangle,
Similarity, Trisection of Sides.
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27. |
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Proposed Problem
123. Area of triangle,
Similarity, Trisection of Sides.
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26. |
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Proposed Problem
122. Marion Walter's Theorem.
Area of triangle and Hexagon, Trisection of Sides.
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25. |
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Proposed Problem
121. Similarity and Area
of triangle, Trisection of Sides.
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24. |
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Proposed Problem
103.Equilateral Triangle Area,
Interior Point, Heron's Formula.
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23. |
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Proposed Problem
68: Triangle, Incircle, Inradius, Tangent, Similarity.
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22. |
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Proposed Problem
36.
Right
Triangle, altitude, incircles and inradii. |
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21. |
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Proposed Problem 24
Right
Triangle, altitude, incircles and inradii. |
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20. |
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Proposed Problem 22
Right
Triangle, altitudes, incircles and inradii. |
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19. |
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Proposed problem 20
Right
Triangle, incircles and inradii. |
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18. |
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Proposed Problem
66: Triangle, Excircle, Tangents, Geometric Mean.
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17. |
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Proposed Problem
61: Triangle, Trisection of Sides |
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16. |
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Archimedes Proposition 7.
Circles and Squares |
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15. |
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Lunes of
Hippocrates 4
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14. |
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Lunes of
Hippocrates 3
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13. |
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Lunes of
Hippocrates 2
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12. |
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Lunes
of Hippocrates 1
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11. |
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Pythagoras and Circle Area/a>
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10. |
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Sangaku Geometry Theorem: Three circles and a tangent line. |
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9. |
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Proposed Problem
98/a>.. Quadrilateral Area, Centroid,
Similarity. |
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8. |
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Proposed Problem
97/a>.. Similar Triangles, Areas. |
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7. |
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Proposed Problem
96/a>.. Similar Triangles, Incenters, Parallelogram. |
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6. |
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Proposed Problem
95/a>.. Similar Triangles, Inradii,
Parallel. |
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5. |
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Proposed Problem
94/a>.. Similar Triangles, Circumcircles,
Circumradii. |
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4. |
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Proposed Problem
93/a>.. Similar Triangles, Circumcircles, Parallelogram. |
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3. |
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Proposed Problem
92/a>.. Similar Triangles, Circumcircles, Circumradii, Parallel. |
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2. |
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Proposed Problem
91/a>.. Similar Triangles, Altitude, Parallel. |
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1. |
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Triangle and Squares 17/a>
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