Online Geometry: Perpendicular Lines, Theorems and Problems

Perpendicular Lines, Theorems and Problems - Table of Content

Points, Theorems and Problems - Index.
Intersection Point.

Perpendicular Bisector.

Triangle Circumcenter by Folding, Video.
Dynamic Geometry Software: Tabula by Numeracy Works.
Step-by-Step construction, Manipulation, and animation.
Perpendicular Bisectors.

Rectangle. Theorems and Problems Index.

Geometry Problem 721.
Triangle, Orthocenter, Circumcircle, Altitude, Collinear points.

Geometry Problem 713.
Tangent Circles, Diameter, Chord, Tangent, Incenter, Incircle, Triangle.

Geometry Problem 712.
Semicircle, Diameter, Tangent, Perpendicular, Angle Measure.

Geometry Problem 705.
Intersecting Circles, Diameter, Angles, Measurement, Perpendicular.

Geometry Problem 704.
Triangle, Orthocenter, Altitude, Midpoint, Perpendicular.

Geometry Problem 701.
Intersecting Circles, Diameter, Perpendicular, Angles, Congruence.

Geometry Problem 680.
Concentric Circles, Radii, Chords, Perpendicular, Metric Relations.

Geometry Problem 678.
Triangle, Simson Line, Circumcircle, Tangent, Parallel, Perpendicular, Collinear Points.

Geometry Problem 660.
Triangle, Circle, Diameter, Perpendicular Lines.

Geometry Problem 658.
Triangle, Altitudes, Diameter, Circle, Angles.

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

Geometry Problem 651.
Archimedes Book of Lemmas: Proposition 11
Perpendicular chords, Arcs, Radius.

Geometry Problem 650.
Tangent Circles, Diameter, Radius, Chord, Perpendicular.

Geometry Problem 649.
Archimedes Book of Lemmas: Proposition 10
Tangent, Chord, Parallel, Midpoint.

Geometry Problem 648.
Archimedes Book of Lemmas: Proposition 9
Perpendicular chords, Arcs.

Geometry Problem 639
Semicircle, Diameter, Perpendicular, Inscribed Circle, Tangent, Arbelos, Congruence.

Geometry Problem 637
Semicircle, Diameter, Perpendicular, Inscribed Circle, Chord, Tangent, Arbelos.

Geometry Problem 636
Semicircle, Diameter, Perpendicular, Inscribed Circle, Common Tangent.

Geometry Problem 635
Semicircle, Diameter, Perpendicular, Inscribed Circle, Radius.

Geometry Problem 629
Triangle, Perpendicular to Sides, Intersecting Lines, Altitude.

Geometry Problem 613
Altitude of a Triangle, Perpendicular, Collinear Points.

Geometry Problem 612
Altitude of a Triangle, Perpendicular, Congruence.

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

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

Geometry Problem 609
Triangle, Perpendiculars, 90 Degrees, Collinearity.

Problem 606
Isosceles Triangles, Perpendicular and Parallel Lines.

Geometry Problem 589.
Triangle, Incenter, Incircle, Tangency Points, Midpoint, Perpendicular, Distance.

Problem 578.
Circle, Perpendicular Chords, Midpoints, Curvilinear Triangle, Area, 90 Degrees.

Proposed Problem 437.
Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.

Proposed Problem 436.
Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.

Proposed Problem 430.
Circumscribed and Inscribed Regular Pentagon, Perpendicular, Areas.

Proposed Problem 429.
Circumscribed and Inscribed Regular Pentagon, Perpendicular, Measurement.

Proposed Problem 428.
Quadrant of a circle, Square, Angle bisector, Measurement.

Proposed Problem 427.
Triangle, Two Altitudes, Square of a Side.

Proposed Problem 420.
Triangle, Altitude, Angles, Sides, Measurement.

Proposed Problem 414.
Triangle, Angles, Altitude, Median, Congruence.

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

Proposed Problem 409.
Quadrilateral, Diagonals, Perpendicular, Congruence.

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

Proposed Problem 397.
Triangle, Altitude, Midpoints, Congruence.

Proposed Problem 396.
Square, Angle Trisectors, Congruence, Area.

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

Proposed Problem 387.
Triangle, Angle bisector, Perpendicular, Concyclic points.

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 377.
Triangle, Internal Angle Bisector, Altitude.

Proposed Problem 373.
Square, Inscribed Circle, Diagonal, Perpendicular, Angle.

Proposed Problem 368.
Triangle, 120 degrees, Angle bisectors, Perpendicular.

Proposed Problem 340.
Triangle, Angle Bisectors, Perpendiculars, Distances. Exterior Point.

Proposed Problem 339.
Triangle, Angle Bisectors, Perpendiculars, Distances. Interior Point.

Proposed Problem 335.
Cyclic Quadrilateral, Perpendiculars to Sides.

Proposed Problem 334.
Cyclic Quadrilateral, Perpendiculars to Diagonals.

Proposed Problem 333.
Circle inscribed in a semicircle, Perpendicular to the common tangent.

Proposed Problem 330.
Cyclic quadrilateral, Perpendicular diagonals, Area, Circumcenter.

Proposed Problem 329.
Triangle, Altitudes, Circle, Diameter, Concyclic points.

Proposed Problem 324.
Quadrilateral with perpendicular diagonals, Concurrence.

Proposed Problem 319.
Triangle, Altitudes, Perpendiculars, Collinear points.

Proposed Problem 318.
A Scalene triangle, Altitudes, Collinear points.

Proposed Problem 317.
Right triangle and Inscribed Squares.

Proposed Problem 316.
Circular segments and Inscribed Squares.

Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square.

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

Proposed Problem 306.
Square, Triangles, Angles, Sides.

Proposed Problem 305.
Square, Triangles, Angles, Sides.

Proposed Problem 303.
Triangle, Angle bisector of 90 degrees.

Proposed Problem 297.
Intersecting Circles, Chord, Secant, Radius, Angle, Perpendicular.

Proposed Problem 296.
Intersecting Circles, Chord, Radius, Angle, Perpendicular.

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

Proposed Problem 295.
Archimedean Twin Circles, Arbelos, Semicircles, Harmonic Mean, Radii, Perpendicular lines.

Proposed Problem 294.
Right triangle, Circumcenter, Excenter, Hypotenuse, Perpendicular lines.

Proposed Problem 291.
Triangle, Circle, Circumradius, Perpendicular lines.

Archimedes' Book of Lemmas Proposition 2
Semicircle, tangents, secant line, perpendicular line to diameter. 

Archimedes' Book of Lemmas Proposition 3
Semicircle, perpendicular line to diameter, chords.

Archimedes' Book of Lemmas Proposition 4
Semicircles, perpendicular line to diameter, Arbelos.

Proposition 5
Arbelos, perpendicular line to diameter, Archimedean Twins, Semicircles, Circles

Proposition 9.
Perpendicular chords, sum of arcs.

Proposition 9.
Perpendicular chords, sum of arcs.

Proposition 10
Tangents to a circle, secant line and chords, Parallel.

Proposition 11
 Perpendicular chords  and radius, square.

Proposition 12
Semicircle, Perpendicular to diameter, Chords, Tangents.

Proposition 13
Circle, Diameter, Perpendicular to chords, Congruence.

Proposition 14
Salinon: Semicircles, Perpendicular to diameter, Circles, Areas.

Proposition 15
Circle, Perpendicular to diameter, Inscribed Regular Pentagon, Arc, Midpoint,  Radius.

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

Proposed Problem 290.
Internally Tangent circles, Radius, Perpendicular lines, Tangent.

Geometry Expressions.
Perpendicular lines

Proposed Problem 289: Tangent circles, Radius, Perpendicular, Tangent.

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

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

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

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

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

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

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

Eight-Point Circle Theorem
Step-by-Step construction, Manipulation, and animation.
TracenPoche Interactive Geometry Software
Perpendicular diagonals.

Simson Line. using Tracenpoche Interactive Geometry Software.
Step-by-Step construction, Manipulation, and animation.
Perpendicular diagonals.

Triangle: Incircle, Perpendicular, Angle Bisector.
Using Tracenpoche Interactive Geometry Software.
Step-by-Step construction, Manipulation, and animation
Perpendicular diagonals.

Orthopole of a Triangle
Using TracenPoche Dynamic Geometry Software
Perpendicular diagonals.

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, Perpendicular lines, Parallel lines.  

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

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

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

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

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

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

Proposed Problem 207. Right Triangle, Hypotenuse, Inradius, Exradius relative to the hypotenuse, Perpendicular lines.  

Proposed Problem 206. Area of a Right Triangle, Inradius, and Exradius relative to the hypotenuse, Perpendicular lines.  

Proposed Problem 205. Right Triangle Area, Exradii relatives to legs or catheti, Perpendicular lines.  

Proposed Problem 204. Right Triangle, Incircle, Excircles, Inradius, Exradii, Perpendicular lines.  

Proposed Problem 203. Right Triangle, Excircles, Exradii, Hypotenuse, Perpendicular lines.

Proposed Problem 202. Right Triangle, Incicrle, Excircles relatives to catheti, Points of Tangency, Exradius, Semiperimeter, Perpendicular lines.  

Proposed Problem 201. Right Triangle, Excircles, Points of Tangency, Exradius, Semiperimeter, Perpendicular lines.  

Proposed Problem 200. Right Triangle, Incircle, Excircles, Points of Tangency, Inradius, Perpendicular lines.

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

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

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

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

Proposed Problem 136. Orthic Triangle, Altitudes, Perpendicular lines, Concyclic Points.  

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

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

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

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

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

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

Taylor Circle Theorem Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation, Perpendicular 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 76: Area of a Circle. Square, Circle, Circular Sector, Perpendicular lines.

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

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

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

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

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

Triangle and Squares 22
Perpendicular lines, Parallel lines. 

Triangle and Squares 20
Perpendicular lines, Parallel lines.

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

Triangle and Squares 17, Perpendicular lines.

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

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

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

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

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

Parallelogram with Squares theorem Thébault's Theorem.
Perpendicular lines 

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.
 

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.