Online Geometry: Parallel Lines, Theorems and Problems

Parallelograms. Theorems and Problems.
It is a quadrilateral whose opposite sides are parallel.

Rectangle. Theorems and Problems Index.

Rhombus.

Drawing parallels in geometry.
Mikhail Gromov has been awarded the Abel Prize for his work making sense of the range of geometries that appeared in the 19th century and he is a worthy successor to the likes of Euclid, Riemann and Gauss, Bolyai and Lobachevsky.

Geometry Problem 715.
Triangle, Altitudes, Parallel, Circumcircle, Angle, Measurement.

Geometry Problem 714.
Triangle, Parallel, Cevians, Midpoints, Median, Ceva Theorem.

Geometry Problem 710.
Parallelogram, Midpoint, Diagonal, Collinear Points, Metric Relations, Triangle, Similarity, Congruence.

Geometry Problem 709.
Parallelogram, Midpoint, Diagonal, Metric Relations, Triangle.

Geometry Problem 702.
Triangle, Circumcircle, Parallel, Concyclic Points.

Geometry Problem 659.
Parallelogram, Parallel Lines, Collinear Points.

Geometry Problem 657.
Triangle, Incircle, Tangency Points, Parallel Lines, Congruence.

Geometry Problem 633
Triangle, Parallel Lines, Intersecting Lines.

Geometry Problem 628
Cyclic Quadrilateral, Congruence, Parallel Lines.

Geometry Problem 625
Triangle, Parallel Lines, Congruence, Angle Bisector.

Geometry Problem 621
Three Rectangles, Diagonals, Centers, Angles.

Geometry Problem 616
Parallelogram, Diagonals, Intersecting Lines, Parallel Lines.

Geometry Problem 614
Parallelogram, Diagonals, Intersecting Lines, Parallel Lines.

Geometry Problem 611
Altitude of a Triangle, Perpendicular, Collinear Points, Parallel.

Geometry Problem 610
Intersecting Circles, Radius, Parallel, Common Tangent, Angle, 90 Degrees.

Problem 606
Isosceles Triangles, Perpendicular and Parallel Lines.

Problem 531.
Triangle, Angle Bisector, Midpoint, Parallel, Perimeter, Semiperimeter.

Problem 518.
Parallelogram, Five Squares, Centers, 45 Degrees, Measure.

Problem 494.
Circular Sector, 90 Degrees, Semicircle, Chord, Parallel.

Problem 492.
Cyclic Quadrilateral, Circle, Perpendicular, Side, Diagonal, Parallel.

Problem 489.
Parallelogram, Triangle, Quadrilateral, Area.

Problem 487.
Intersecting Circles, Area, Diameter, Parallel, 90 Degrees.

Problem 477.
Parallelogram, Triangle, Quadrilateral, Area.

Problem 475.
Tangent Circles, Secant, Tangent, Parallel.

Problem 474.
Parallelogram, Diagonal, Circle, Vertex.

Problem 473.
Triangle, Parallel, Side, Parallelogram, Area, Similarity.

Problem 472.
Triangle, Parallel, Side, Parallelogram, Area, Similarity.

Problem 471.
Triangle, Parallel, Side, Concurrency, Area, Similarity.

Problem 470.
Triangle, Parallel, Side, Parallelogram, Area, Similarity.

Proposed Problem 438.
Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement, Parallel lines.

Proposed Problem 432.
Trapezoid, Parallel, Measurement, Similarity, Transversal.

Proposed Problem 413.
Cyclic Quadrilateral, Orthocenter, Parallelogram, Concurrency, Congruence.

Proposed Problem 408.
Cyclic quadrilateral, Perpendicular, Parallelogram, Congruence.

Proposed Problem 402.
Hexagon, Triangle, Centroid, Parallel, Congruence, Similarity.

Proposed Problem 398.
Parallelogram, Area, Similarity, Congruence.

Proposed Problem 392.
Triangle, Parallel lines, Concurrent lines.

Proposed Problem 391.
Triangle, Parallel lines, Collinear points.

Proposed Problem 390.
Triangle, Parallel lines, Collinear points.

Proposed Problem 389.
Triangle, Parallel lines, Harmonic Mean.

Proposed Problem 388.
Triangle, Angle bisector, Perpendicular, Parallel.

Proposed Problem 386.
Triangle, Angle bisector, Perpendicular, Parallel, Semiperimeter, Congruence.

Proposed Problem 385.
Triangle, Angle bisector, Perpendicular, Parallel, Semiperimeter.

Proposed Problem 384.
Triangle, Angle bisector, Perpendicular, Parallel, Semiperimeter.

Proposed Problem 380.
Triangle, Excenter, Parallel to a side, Angle, Congruence.

Proposed Problem 379.
Triangle, Excenter, Parallel to a side, Angle, Congruence.

Proposed Problem 378.
Triangle, Incenter, Parallel to a side, Angle, Congruence.

Proposed Problem 337.
Isosceles Trapezoid, Angle bisector, Parallel, Concyclic points.

Proposed Problem 328.
Triangle, Incircle, Tangency Points, Parallel, Midpoint.

Proposed Problem 316.
Circular segments and Inscribed Squares.
Parallel lines.

Proposed Problem 313.
Circle, Chord, Tangent, Perpendicular, Geometric Mean.
Parallel lines.

Proposed Problem 293.
Inscribed Quadrilateral, Perpendicular, Rectangle, Isosceles Right triangle, Area, Similarity, Parallel lines.

Proposed Problem 292.
Triangle, Parallel, Congruence, Similarity.

Archimedes' Book of Lemmas Proposition 1.
Tangent circles and parallel diameters 

Archimedes' Book of Lemmas Proposition 10
Tangents to a circle, secant line and chords, Parallel lines.

Archimedes' Book of Lemmas Proposition 13
Circle, Diameter, Chord, Perpendicular, Congruence, Parallel lines.

Hyperbolic Tessellation and Geometric Art.
Hyperbolic geometry is a non-Euclidean geometry in which it is assumed that through any point there are two or more parallel lines that do not intersect a given line in the plane. 

Proposed Problem 292.
Triangle, Parallel lines, Congruence, Similarity.

Euclid's Elements, Book X, Lemma for Proposition 33
One page visual illustration. Parallel lines.

Engineering the Impossible: Chartres Cathedral
Circle, Square, Triangle, Parallel lines.

Proposed Problem 280.
Quadrilateral, Perpendiculars, , Parallel lines, Area of Squares.

IC 4406: A Seemingly Square Nebula,
Evidence indicates that IC 4406 is likely a hollow cylinder, with its square appearance the result of our vantage point in viewing the cylinder from the side.
Credit: C. R. O'Dell (Vanderbilt U.) et al., Hubble Heritage Team, NASA.
Parallel lines.

Piet Mondrian, New York City 1941-42 and the Golden Rectangle.
Index.
Parallel lines.

Piet Mondrian, Lozenge Composition with Red, Black, Blue, and Yellow 1925,
Pioneer of Geometric abstraction.

Proposed Problem 273.
Triangle, Perpendiculars, Parallel lines, Area of Squares.

Proposed Problem 252.
Rectangle, isosceles right triangle, square, area.

Proposed Problem 251.
Parallelogram, sides, diagonals, squares, areas.

Proposed Problem 245. Parallelogram with Equilateral triangles

Proposed Problem 237. Quadrilateral, Midpoints, Exterior line, Perpendicular lines, Parallel lines.

Proposed Problem 236. Quadrilateral, Midpoints, Exterior line, Perpendicular lines, Parallel lines.

Proposed Problem 235. Parallelogram, Transversal line, Perpendicular lines, Parallel lines.  

Proposed Problem 234. Parallelogram, Line through a vertex, Perpendicular lines, Parallel lines.

Proposed Problem 233. Parallelogram, Exterior line, Perpendicular lines, Parallel lines.

Proposed Problem 232. Parallelogram, Line through a vertex, Perpendicular lines.

Proposed Problem 231. Triangle, Midpoints, Transversal, Perpendicular lines, Parallel lines.

Proposed Problem 230. Triangle, Midpoints, Transversal, Perpendicular lines.

Proposed Problem 229. Triangle, Centroid, Transversal, Perpendicular lines, Parallel lines.

Proposed Problem 228. Triangle, Midpoints, Exterior line, Perpendiculars lines.

Proposed Problem 227. Triangle, Centroid, Exterior line, Perpendicular lines, Parallel lines.  

Proposed Problem 226. Triangle, Centroid, Perpendicular lines, Parallel lines.  

Proposed Problem 224. Viviani's theorem, Isosceles triangle, Altitude, Distances, Point on the extension of the base, Parallel lines.  

Proposed Problem 223. Viviani's theorem, Isosceles triangle, Altitude, Distances, Parallel lines.  

Proposed Problem 222. Viviani's theorem, Equilateral triangle, Exterior point, Distances.  

Proposed Problem 221. Viviani's theorem, Equilateral triangle, Interior point, Distances, Parallel lines.  

Proposed Problem 219. Rhombus, perpendiculars, arithmetic mean, Parallel lines.

Proposed Problem 218. Right triangle, Altitude and Projections, Parallel lines.

Proposed Problem 217. Right triangle, Altitude and Projections, Parallel lines.

Archimedes Arbelos and Square 2. Dynamic Geometry Software.
Step-by-Step construction, Manipulation, and animation, Parallel lines.

Archimedes Arbelos and Square 1. Dynamic Geometry Software.
Step-by-Step construction, Manipulation, and animation, Parallel lines.

Proposed Problem 185. Trapezoid, Triangles and Angles, Parallel lines.  

Proposed Problem 179. Rhombus, Square, Diagonal, Angle, Parallel lines.  

Proposed Problem 177. Parallelogram with Midpoints, Triangles, Areas, Parallel lines.  

Proposed Problem 172. Trapezoid, Midpoints, Quadrilaterals, Areas, Parallel lines.  

Proposed Problem 171. Trapezoid, Midpoints, Triangles, Areas, Parallel lines.  

Proposed Problem 170. Trapezoid, Midpoint, Triangle, Area, Parallel lines.  

Proposed Problem 169. Parallelogram, Interior and Exterior Points, Diagonals, Pentagon, Triangles, Areas, Parallel lines.  

Proposed Problem 168. Parallelogram, Diagonal, Triangles, Pentagon, Areas, Parallel lines.  

Proposed Problem 167. Parallelogram, Diagonal, Triangles, Areas, Parallel lines.  

Proposed Problem 166. Parallelogram, Diagonal, Triangles, Areas, Parallel lines.  

Proposed Problem 165. Parallelogram, Diagonal, Triangles, Areas, Parallel lines.  

Proposed Problem 164. Parallelogram, Trapezoid, Diagonal, Triangles, Areas, Parallel lines.  

Proposed Problem 163. Trapezoid, Diagonals, Triangles, Areas, Parallel lines.  
Trapezoid Index

Proposed Problem 162. Parallelogram, Triangles, Areas, Parallel lines.  

Proposed Problem 161. Parallelogram, Midpoints, Octagon, Areas, Parallel lines.  

Proposed Problem 146. Varignon's Theorem: Quadrilateral, Midpoints, Parallelogram, Area, Perimeter, Parallel lines.

Proposed Problem 145. Four Triangles, Incircle, Tangent and Parallel to Side, Incenters, Circumcenters.  

Proposed Problem 144. Four Triangles, Incircle, Tangent and Parallel to Side, Inradii.

Proposed Problem 143. Four Triangles, Incircle, Tangent and Parallel to Side, Circumradii.

Proposed Problem 142. Four Triangles, Incircle, Tangent and Parallel to Side, Areas.

Proposed Problem 141. Triangle, Incircle, Tangent , Parallel, Perimeters.

Proposed Problem 137. Orthic Triangle, Altitudes, Perpendicular, Incircle, Collinear Points, Parallelogram.  

Proposed Problem 135. Orthic Triangle, Altitudes, Perpendicular, Parallel.  

Trapezoid, Triangle, Diagonals, Midpoints. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Parallel lines. Prove proposition.

Proposed Problem 127. Centroid and Incenter of a Triangle, Parallel lines, Proportions.

Euclid's Elements Book VI, Proposition 3: Angle Bisector Theorem
Parallel lines.

Bottema's Theorem:
Triangle and Squares with Interactive Geometry Software
Step-by-Step construction, Manipulation, and animation.
Parallel lines.

Proposed Problem 112. Area of square and triangle.
Parallel lines.

The Pythagorean Curiosity Triangles and squares, fifteen conclusions.
Parallel lines.

Triangle and Squares Theorem 1: with Interactive Geometry Software
Step-by-Step construction, Manipulation, and animation.
Parallel lines.

Proposed Problem 98. Quadrilateral Area, Centroid, Similarity.
Parallel lines.

Proposed Problem 97. Similar Triangles, Areas, Parallel lines.

Proposed Problem 96. Similar Triangles, Incenters, Parallelogram, Parallel lines.

Proposed Problem 95. Similar Triangles, Inradii, Parallel lines.

Proposed Problem 94. Similar Triangles, Circumcircles, Circumradii, Parallel lines.

Proposed Problem 93. Similar Triangles, Circumcircles, Parallelogram, Parallel lines.

Proposed Problem 92. Similar Triangles, Circumcircles, Circumradii, Parallel, Parallel lines.

Proposed Problem 91. Similar Triangles, Altitude, Parallel lines.

Taylor Circle Theorem Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation, Parallel lines.
Henry Martyn Taylor and the blind student of mathematics

Proposed Problem 78: Angles of a Circle. Perpendicular and parallel lines, Midpoint, Diameter, Chord, Cyclic quadrilateral, Congruence.

Proposed Problem 77: Angles of a Circle. Parallel lines, Cyclic quadrilateral.

Proposed Problem 76: Area of a Circle. Square, Circle, Circular Sector, Parallel lines.

Interactive Orthopole of a Triangle: Using TracenPoche Dynamic Geometry Software, Online Step-by-Step construction, manipulation, and animation, Parallel lines.

Proposed Problem 72: Intersecting Circles. Cyclic quadrilateral, Chords, Parallel lines.

Proposed Problem 70: Squares inscribed in a triangle, Similarity.
Parallel lines.

Proposed Problem 69: Square inscribed in a triangle, Similarity.
Parallel lines.

Proposed Problem 62: Square Diagonal, Inscribed Circle, Parallel lines.

Sangaku Geometry Theorem: Cyclic Quadrilateral. Inradii, Rectangle, Parallel lines.

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.
Parallel lines.

Triangle and Squares 22
Parallel lines. 

Triangle and Squares 20
Parallel lines.

News: Infants Are Able To Detect The 'Impossible' At An Early Age

Proposed Problem 49. Angles and right triangle, Parallel lines.

Triangle and Squares 17, Parallel lines.

Problem 39. Triangle, Incircle, Bisector, Cyclic Quadrilateral and angles, Parallel lines.

Proposed Problem 29.
Right Triangle, altitude, incircle and inradius, Parallel lines. Ten conclusions.

Heron's Formula with medians 
Parallel lines.

Proposed Problem 27
Right Triangle, incircles and inradius, Parallel lines.

Eyeball Theorem: Animated Angle to Geometry Study.
Parallel lines.

Proposed problem 20
Right Triangle, Parallel lines, incircles and inradii.

Parallelogram with Squares theorem Thébault's Theorem.
 

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.
 

Varignon and Wittenbauer paralellograms. Quadrilateral: midpoints and trisection points of the edges, Parallel lines.

Euclid's Elements Book I, 23 Definitions. One-page visual illustration.
Euclid's Elements Book. Index

Geometry Online Glossary . Octagon.
Geometry Glossary based on the new New York State mathematics standards initiative.

Interactive Geometry Glossary using AnswerTips technology.
Parallel lines.

Cuzco (Cusco, Qusqu). Sacsayhuaman, Inca fortress, was constructed from gigantic stones. Three massive parallel walls, built in a zig-zag pattern, were part of the defensive fortress.

The Incas and Pottery.
Inca designs were nearly always geometrical and conventional. They included squares repeated one within the other, cross-hatching, rows of triangles, parallel lines, rows of lozenges,