Quadrilaterals: Theorems and Problems
6
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![Quadrilateral Index](../geometry/quadrilateral_index_170.jpg) |
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![Trapezoid, Triangle, Area](../problem/p170_trapezoid_triangle_17.jpg) |
Proposed Problem 170. Trapezoid,
Midpoint, Triangle, Area.
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![Geometry Problem: Parallelogram, Triangles, Pentagon, Area](../problem/p169_parallelogram_area_17.jpg) |
Proposed Problem 169.
Parallelogram, Interior and Exterior Points, Diagonals, Pentagon,
Triangles, Areas
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![Geometry Problem: Parallelogram, Triangles, Pentagon, Area](../problem/p168_parallelogram_area_17.jpg) |
Proposed Problem 168.
Parallelogram, Diagonal, Triangles, Pentagon, Areas.
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![Geometry Problem: Parallelogram, Triangles, Area](../problem/p167_parallelogram_area_17.jpg) |
Proposed Problem 167.
Parallelogram, Diagonal, Triangles, Areas.
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![Geometry Problem: Parallelogram, Triangles, Area](../problem/p166_parallelogram_area_17.jpg) |
Proposed Problem 166.
Parallelogram, Diagonal, Triangles, Areas.
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![Elearning 165: Parallelogram area](../problem/p165_parallelogram_area_17.jpg) |
Proposed Problem 165.
Parallelogram, Diagonal, Triangles, Areas.
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![Elearning 164: Parallelogram, Triangles, Area](../problem/p164_parallelogram_area_17.jpg) |
Proposed Problem 164.
Parallelogram, Trapezoid, Diagonal, Triangles, Areas.
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![Elearning 163 Trapezoid, Triangles, Area](../problem/p163_trapezoid_triangle_17.jpg) |
Proposed Problem 163. Trapezoid,
Diagonals, Triangles, Areas.
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![Elearning 162 Parallelogram and Triangles Areas](../problem/p162_parallelogram_area_17.jpg) |
Proposed Problem 162.
Parallelogram, Triangles, Areas.
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![Elearning 161 Parallelogram and Octagon Areas](../problem/p161_parallelogram_area_17.jpg) |
Proposed Problem 161.
Parallelogram, Midpoints, Octagon, Areas.
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![Circumscribed quadrilateral](../problem/p153_quadrilateral_circle_1.jpg) |
Proposed Problem 153.
Circumscribed Quadrilateral, Diagonals Concurrent with Chords.
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![Circumscribed Quadrilateral](../problem/p152_circumscribed_17.jpg) |
Proposed Problem 152.
Circumscribed Quadrilateral, Diagonal, Chord, Proportion.
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![Quadrilateral area, trisection](../problem/p151_quadrilateral_area_17.jpg) |
Proposed Problem 151.
Quadrilateral, Area, Trisection of Sides.
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![Machu Picchu](../wonder_world/golden_machu_picchu_17.jpg) |
Machu Picchu and Golden Rectangle.
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![Quadrilateral area, trisection](../problem/p150_quadrilateral_area_17.jpg) |
Proposed Problem 150.
Quadrilateral, Area, Trisection of Sides.
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![Quadrilateral Area, Midpoint, Elearning](../problem/p149_quadrilateral_area_17.jpg) |
Proposed Problem 149.
Quadrilateral, Area, Midpoints.
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![Quadrilateral area, Midpoint, Elearning](../problem/p148_quadrilateral_area_17.jpg) |
Proposed Problem 148.
Quadrilateral, Area, Midpoints.
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![Quadrilateral Area, Elearning](../problem/p147_quadrilateral_area_17.jpg) |
Proposed Problem 147.
Quadrilateral, Area, Midpoints, Triangle.
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![Quadrilateral Area, Midpoint](../problem/p146_quadrilateral_area_17.jpg) |
Proposed Problem 146. Varignon's
Theorem: Quadrilateral, Midpoints, Parallelogram, Area, Perimeter. |
![Elearning 139: Triangle Area](../problem/p139_area_triangle_17.jpg) |
Proposed Problem 139. Triangle
Area, Orthic Triangle, Semiperimeter, Circumradius. |
![Geometry Problem 137](../problem/p137_orthic_triangle_17.jpg) |
Proposed Problem 137.
Orthic Triangle, Altitudes, Perpendicular, Incircle, Collinear Points,
Parallelogram. |
![Trapezoid](../geometry/trapezoid_diagonal_01_17.jpg) |
Trapezoid, Triangle, Diagonals, Midpoints.
Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Prove proposition. |
![](../geometry/four_circles_concyclic_17.jpg) |
Four Circles Theorem Using Interactive Dynamic Software
Step-by-Step construction, Manipulation, and animation.
Cyclic Quadrilateral. |
![Pythagoras curiosity. Online tutoring.](../pythagoras/pythagoras_curiosity_17.jpg) |
The Pythagorean Curiosity
Triangles and squares, fifteen conclusions |
![Inca Skull Surgery](../incas/inca_skull_surgeon_17.jpg) |
Inca Skull Surgery
Inca Skull Surgeons Were "Highly Skilled," Study Finds.
Trepanation = square hole in the skull. |
![Triangle Square Theorem 1. Elearning, Online Tutoring](../geometry/triangle_square_01_17.jpg) |
Triangle and Squares Theorem 1: with Interactive Geometry
Software
Step-by-Step construction, Manipulation, and animation. |
![Bottema Theorem. Elearning.](../geometry/bottema_theorem_elearning.jpg) |
Bottema's Theorem: Triangle and Squares with Interactive
Geometry Software
Step-by-Step construction, Manipulation, and animation. |
![Eight Point Circle Theorem. Elearning](../geometry/eight_point_circle_17.jpg) |
Eight-Point Circle Theorem
Step-by-Step construction, Manipulation, and animation. |
![Complete quadrilateral: ortholine, Elearning](../geometry/quadrilateral_ortholine_17.jpg) |
Complete Quadrilateral: Ortholine-Steiner Line.
Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation |
![Elearning 96](../problem/p096_parallelogram_elearning_17.jpg) |
Parallelogram Problem 96. Similar
Triangles, Incenters, Parallelogram. |
![Elearning 93 Parallelogram](../problem/p093_triangle_elearning_17.jpg) |
Parallelogram Problem 93. Similar
Triangles, Circumcircles, Parallelogram. |
![Geometry problem](../problem/p090_quadrilateral_area_17.jpg) |
Proposed Problem 90.
Quadrilateral and Triangle Areas, Midpoints.
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![Math problem](../problem/p088_quadrilateral_area_17.jpg) |
Proposed Problem 88. Triangle and Quadrilateral Areas, Midpoints.
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![Quadrilateral area problem](../problem/p087_quadrilateral_area_17.jpg) |
Proposed Problem 87. Area of
Quadrilaterals and Midpoints of Diagonals.
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![](../geometry/newton_theorem_line_17.jpg) |
Complete quadrilateral
Newton's Theorem, Newton-Gauss Line: Complete quadrilateral theorem.
Using TracenPoche Dynamic Geometry Software,
Online Step-by-Step construction, manipulation, and animation. |
![](../problem/p078_circle_angles_17.jpg) |
Cyclic quadrilateral
Proposed Problem 78: Angles of a Circle.
Perpendicular and parallel lines, Midpoint, Diameter, Chord, Cyclic
quadrilateral, Congruence.
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![Math Problem 77 Angles](../problem/p077_circle_angle_cyclic_17.jpg) |
Cyclic quadrilateral
Proposed Problem 77: Angles of a Circle.
Parallel lines, Cyclic quadrilateral.
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![](../problem/p076_square_circle_17.jpg) |
Square
Proposed
Problem 76: Area of a Circle.
Square, Circle, Circular Sector. |
![](../problem/p074_intersecting_circles_1.jpg) |
Proposed Problem 74: Three Intersecting Circles.
Cyclic quadrilateral, Angles.
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![](../problem/p073_intersecting_circles_1.jpg) |
Proposed Problem 73: Three Intersecting Circles.
Cyclic quadrilateral.
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![](../problem/p072_circle_secant_17.jpg) |
Proposed Problem 72: Intersecting Circles.
Cyclic quadrilateral, Chords, Parallel.
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![](../problem/p071_cyclic_quadrilateral_1.jpg) |
Proposed Problem 71: Cyclic Quadrilateral.
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![](../problem/p070_squares_inscribed_17.jpg) |
Proposed Problem 70: Squares inscribed in a triangle, Similarity.
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![](../problem/p069_square_inscribed_17.jpg) |
Proposed Problem 69: Square inscribed in a triangle, Similarity.
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![](../problem/p067_triangle_circumcircle_.jpg) |
Proposed Problem 67: Triangle, Circumcircle, Angles, Cyclic
Quadrilateral.
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![](../geometry/four_circles_concyclic_17.jpg) |
Four Circles Theorem Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation. |
![](../problem/p063_regular_heptagon_17.jpg) |
Proposed Problem 63: Heptagon
Regular, Side and Diagonals.
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![](../problem/p062_square_diagonal_17.jpg) |
Proposed Problem 62: Square
Diagonal, Inscribed Circle.
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Arbelos, Inscribed Circle and Concyclic Points
Cyclic Quadrilateral. |
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Casey's Theorem. Generalized Ptolemy's Theorem. |
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Proposed Problem 57: Angle bisector, circles
Cyclic Quadrilateral. |
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Proposed Problem 55: Angle bisector, circles
Cyclic Quadrilateral. |
![](../problem/aubel_villiers_theorem_170.jpg) |
Generalizing Van Aubel: Michael de Villiers' Theorems.
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![](eyeball_theorem_170.jpg) |
Eyeball Theorem:
Animated Angle to Geometry Study.
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![](../pythagoras/squaring_circle_01_170.jpg) |
Hippocrates and Squaring the Circle
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![](../triangle_squares_1_170.jpg) |
Triangle with Squares
Problem
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![](../parallelogramsquare170x100.jpg) |
Thébault's Theorem. Parallelogram with Squares theorem
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![](../Varignon_Wittenbauer_170.jpg) |
Varignon and Wittenbauer parallelograms. Quadrilateral: midpoints and
trisection points of the edges.
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![Van Aubel](../vanaubel170x100.jpg) |
Van Aubel's Theorem.
Quadrilateral with Squares. Proof with animation.
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![](../equilic/cyclic_ptolemy_ratio_170.jpg) |
Cyclic
Quadrilateral: Ratio of the Diagonals
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![](../circle/brahmagupta_theorem_170.jpg) |
Brahmagupta's Theorem
Cyclic quadrilateral.
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![](../circle/brahmagupta_corollary_170.jpg) |
Brahmagupta's Corollary
Cyclic quadrilateral.
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![](../circle/brahmagupta_formula_170.jpg) |
Brahmagupta's Formula
Area of a cyclic quadrilateral.
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![](../circle/brahmagupta_extension_170.jpg) |
Brahmagupta Formula
Extension
Area of any quadrilateral.
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![](../NewtonTheorem170.jpg) |
Newton's
Theorem: Newton's Line. Circumscribed quadrilateral, midpoints of
diagonals, center of the circle inscribed.
Puzzle of the Newton's
Theorem: 50 pieces of circles. |
![](../sangaku_2a_170.jpg) |
Sangaku Problem. The
incenters of four triangles in a cyclic quadrilateral form a rectangle. |
![](../Kurschak170x100.jpg) |
Kurschak's Tile and Theorem.
Jozsef Kurschak (Hungary, 1864-1933) An elegant and a purely geometric
way of finding the area of a regular dodecagon.
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![](../problem/p033_triangle_sat_college_1.jpg) |
Proposed Problem 33.
Triangle and quadrilateral. |
![](../problem/p035_cyclic_quadrilateral_1.jpg) |
Proposed
Problem 35.
Incenters and Inradii in Cyclic Quadrilateral. |
![](../problem/p042_angle_triangle_170.jpg) |
Proposed Problem 42.
Angles and triangles. |
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