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Circles, Theorems and Problems: Table of Content (Page 1 of 5)

 

Circles

 

 

 

Chord Index

Chord. Index.
A chord of a circle is the line segment joining two points on the circle. 

 

Circular Sector Index

Circular Sector. Index.
Theorems and Problems. 

 

Circumcenter Index

Circumcenter. Index.

 

Incenter, Incircle, Index

Incenter of a triangle. Index.

 

Excenter, Excircle, Index

Excenter. Index.
Excircle, Exradius.

 

Soddy Circles and Descartes Theorem

Soddy Circles and Descartes Theorem.
Three tangent circles, Inscribed and Circumscribed Circles, Radii.

 

Macheroni Constructions Index

Mascheroni construction with compass alone. Index

 

Mascheroni Construction with compass alone

Mascheroni construction: Find the center of a circle with compass alone.

 

Mascheroni construction with compass alone

Mascheroni construction: Construct the circumcircle of a triangle with compass alone.

 

Intersecting Circles Index

Intersecting Circles Index.
Theorems and Problems.

 

Scientists: People Walk in Circles

Study: Walking in circles.
Max-Planck-scientists show that people really walk in circles when lost.

 

Tangent Circles Index

Tangent Circles Index
 

 

Cyclic Quadrilateral Index

Cyclic Quadrilateral.
 

 

Tangential or Circumscribed Quadrilateral Index, theorems and problems

Tangential or Circumscribed Quadrilateral
 

 

Euclid's Elements Book I, Proposition 3

Euclid's Elements Book I, Proposition 3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less

 

Euclid's Elements Book I, Proposition 2

Euclid's Elements Book I, Proposition 2: To place at a given point (as an extremity) a straight line equal to a given straight line

 

Euclid's Elements Book I, Proposition 1

Euclid's Elements Book I, Proposition 1: On a given finite line to construct an equilateral triangle

 

Tangential Quadrilateral

Tangential or Circumscribed Quadrilateral: Diagonals, Inradii Theorem.

 

Six Tangential Quadrilaterals Theorem

Six Tangential or Circumscribed Quadrilaterals Theorem.

 

Tangential Quadrilateral: Pitot Theorem

Tangential or Circumscribed Quadrilateral: Pitot Theorem.

 

Triangle, Incircle, Inradius, Tangent

Proposed Problem 418.
Triangle, Incircle, Inradius, Equal Tangent circles, Radius.

 

Tangent circles, Angles

Proposed Problem 417.
Tangent circles, Tangent lines, Angles.

 

Cyclic quadrilateral, Orthocenter, Parallelogram, Congruence

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

 

Cyclic quadrilateral, Perpendicular, Parallelogram, Congruence

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

 

Angle, Circle, Radius, Tangent, Secant, Perpendicular

Proposed Problem 407.
Circle, Angle, Tangent, Perpendicular, Secant, Radius, Center.

 

Circular Sector 90 degrees

Proposed Problem 403.
Circular sector, 90 degrees, Circle, Semicircle, Area.

 

Area of a triangle, Circumcircle, Angle Bisector

Proposed Problem 401.
Area, Triangle, Angle bisector, Circumcircle, Perpendicular bisector, Congruence.

 

Triangle, Angle Bisector, Circumcircle

Proposed Problem 400.
Triangle, Angle bisector, Circumcircle, Perpendicular, Congruence.

 

Square, Circle

Proposed Problem 394.
Square, 90 Degree Arc, Diagonal, Congruence.

 

Triangle, Orthocenter

Proposed Problem 393.
Triangle, Orthocenter, Circumcircles, Congruence, Collinear.

 

Triangle, Excenter, Parallel

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

 

Triangle, Excenter, Parallel

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

 

Triangle, Incenter, Parallel to a side

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

 

Triangle, Altitude, Angle bisector

Proposed Problem 377.
Triangle, Internal Angle Bisector, Altitude.

 

Triangle, Excenter, Internal and External angle bisector,

Proposed Problem 376.
Triangle, Excenter, Internal and External Angle Bisectors.

 

Triangle, Excenter, Angle

Proposed Problem 375.
Triangle, Excenter, External Angle Bisectors.

 

Tiangle, Incenter, Angle

Proposed Problem 374.
Triangle, Incenter, Internal Angle Bisectors.

 

Problem about square and inscribed circle

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

 

Circles, Common tangent

Proposed Problem 372.
Circles, Common Internal and External Tangent, Angles.

 

Square, Triangle, Area, Circle

Proposed Problem 371.
Square, Inscribed circle, Triangle, Area.

 

Triangle with squares, Circumcircles

Proposed Problem 370.
Triangle with squares, Circumcircles, Tangent circles.

 

Intersecting circles, chord, angle, congruence 

Proposed Problem 369.
Intersecting circles, Chord, Center, Angle, Congruence.

 

Equilateral triangle, circumcircle

Proposed Problem 366.
Scalene triangle, Circumcircle, Angles, 60 Degrees, Equilateral triangle.

 

Circular sector 60 degrees

Proposed Problem 365.
Circular Sector of 60 degrees, Midpoints, Perpendicular, Congruence.

 

Triangle area, Circle, Curvilinear triangle

Proposed Problem 364.
Obtuse triangle, Congruence, Circles, Diameter, Curvilinear triangle, Area.

 

Circle, Chord, Perpendicular

Proposed Problem 362.
Circle, Chord, Perpendicular, Equal chords.

 

Right triangle, Incircle, Angle

Proposed Problem 361.
Right triangle, Incircle, Incenter, Tangency points, Angle.

 

Area of Triangle, Circles, Diameters

Proposed Problem 360.
Area of a triangle, Curvilinear triangles, Circles, Diameters.

 

Cyclic quadrilateral, Rhombus

Proposed Problem 359.
Cyclic quadrilateral, Angle bisector, Rhombus.

 

Isosceles triangle 80 80 20

Proposed Problem 358.
Isosceles triangle 80-80-20, Circle, Angles, Congruence.

 

Circles, Tangent, Concyclic points

Proposed Problem 355.
Circles, Common external tangent, Concyclic points.

 

Tangential quadrilateral

Proposed Problem 352.
Tangential quadrilateral, Incircles, Common tangent, Circumscribable or Tangential quadrilateral.

 

Rhombus, Circumscribable quadrilateral

Proposed Problem 351.
Rhombus, Incircles, Common tangent, Circumscribable or Tangential quadrilateral.

 

Triangle, Cevian, Incircles, Angles

Proposed Problem 350.
Triangle, Cevian, Incircles, Tangents, Tangency Point, Angles.

 

Triangle, Cevian, Incircles

Proposed Problem 349.
Triangle, Cevian, Incircles, Tangents, Tangency Point.

 

Common tangents

Proposed Problem 348.
Circles, Common External Tangents, Common Internal Tangent.

 

Triangle, Altitude, Circle, Concyclic points

Proposed Problem 347.
Triangle, Altitude, Perpendicular, Circle, Concyclic points.

 

Geometry Problem: Equal circles, Tangents, Hexagon

Proposed Problem 346.
Equal circles, Tangents, Hexagon, Semiperimeter.

 

Geometry Problem: Equal circles, Tangents

Proposed Problem 345.
Equal circles, Tangents, Concurrent lines, Hexagon, Semiperimeter.

 

Quadrilateral, Circle, Tangent

Proposed Problem 344.
Quadrilateral, Extension of sides, Circle, Tangent, Semiperimeter.

 

Hexagon, Inscribed Circle

Proposed Problem 343.
Hexagon, Inscribed circle, Tangent, Semiperimeter.

 

Pentagon, Circle, Tangent,Semiperimeter

Proposed Problem 342.
Pentagon, Inscribed circle, Tangent, Semiperimeter.

 

Triangle, inscribed circle, tangent

Proposed Problem 341.
Triangle, Inscribed circle, Tangent, Semiperimeter.

 

Problem 338 concyclic points

Proposed Problem 338.
Triangle, Circumcircle, Inscribed Circle, Exterior angle bisector, Concyclic points.

 

Isosceles Trapezoid, Concyclic Points

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

 

Tangent circles, Square

Proposed Problem 336.
Two equal circles, a Common Tangent and a Square.

 

Cyclic Quadrilateral, Perpendiculars to Sides

Proposed Problem 335.
Cyclic Quadrilateral, Perpendiculars to Sides.

 

Problem 334 Cyclic Quadrilateral, Perpendiculars to Diagonals

Proposed Problem 334.
Cyclic Quadrilateral, Perpendiculars to Diagonals.

 

Circle and Inscribed Semicircle

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

 

Cyclic Quadrilateral Problem

Proposed Problem 332.
Cyclic quadrilateral, Ratio of diagonals, Similarity.

 

Elearning 331 Square, Inscribed Circle

Proposed Problem 331.
Square, Point on the Inscribed Circle, Tangency Points.

 

Cyclic quadrilateral

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

 

Triangle, Altitudes, Concyclic Points

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

 

Triangle, Incircle, Parallel, Midpoint

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

 

Right triangle Area, Semicircle, Square

Proposed Problem 327.
Right triangle Area, Incircle, Circumcircle, Square.

 

Equilateral triangle and semicircle

Proposed Problem 326.
Equilateral triangle, Semicircle, Equal arcs.

 

Isosceles triangle, Altitude, Incircle, Excircle

Proposed Problem 325.
Isosceles triangle, Altitude, Cevian, Incircle, Excircle.

 

Triangle, Incenter, Circumcenter, Collinear

Proposed Problem 323.
Triangle, Incenter, Circumcenter, Equal circles, Collinearity.

 

Area of a triangle

Proposed Problem 322.
Square, Inscribed circle, Tangent, Triangle area.

 

Triangle, Circumcenter, Incenter, Excenter, Collinear

Proposed Problem 320.
Triangle, Circumcircle, Incenter, Excenter, Collinear points.

 

Circular Segments, Inscribed Squares

Proposed Problem 316.
Circular segments and Inscribed Squares.

 

Three Tangent Circles

Proposed Problem 315.
Three Tangent circles, Common external tangent line, Geometric Mean.

 

Tangent circles and line

Proposed Problem 314.
Tangent circles, Common external tangent line, Geometric Mean.

 

Circle, Chord, Tangent, Perpendicular

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

 

Elearning 312: Cyclic quadrilateral

Proposed Problem 312.
Cyclic quadrilateral, Side extensions, Tangents.

 

Circle, Semicircle, Chord, Diameter

Proposed Problem 311.
Circle Inscribed in a Semicircle, Chord, Diameter.

 

Circle inscribed in a semicircle

Proposed Problem 310.
Circle Inscribed in a Semicircle, 45 Degrees Angle.

 

Nonagon problem

Proposed Problem 309.
Regular Nonagon or Enneagon, Diagonals, Side.

 

Elearning Geometry Problem 308

Proposed Problem 308.
Regular Nonagon or Enneagon, Midpoints, Side, Arc, Angle.

 

Nonagon, Midpoints, Angle

Proposed Problem 307.
Regular Nonagon or Enneagon, Midpoints, Side, Arc, Angle.

 

Tangents to a circle, Secants

Proposed Problem 301.
Tangents to a circle, Secants, Square.

 

Problem: Tangent to a circle

Proposed Problem 300.
Tangent to a circle, Secants, Square.

 

Intersecting circles, Chord, Secant, Midpoint

Proposed Problem 299.
Intersecting Circles, Chord, Secant, Midpoint, Congruence.

 

Elearning 298: Intersecting circles

Proposed Problem 298.
Intersecting Circles, Chord, Secant, Midpoint, Congruence.

 

Intersecting circles, Chord, Secant, Radius, Angle, Perpendicular

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

 

Intersecting circles

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

 

Archimedes Twin, Harmonic mean

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

 

Right Triangle Circumcenter, Excenter

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

 

Inscribed Quadrilateral

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

 

Triangle, Circle, Radius, Perpendicular

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

 

Tangent Circles, Radius

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

 

Geometry Expressions

Geometry Expressions.

 

Tangent Circles

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

 

Tangent Circles, Harmonic Mean

Proposed Problem 288: Tangent circles, Harmonic Mean, Radius, Diameter.

 

Congruence

Congruence. Index.

 

Similarity

Similarity, Ratios, Proportions. Index.

 

Areas

Areas Index

 

Sagitta, Arc, Chord

Sagitta, Arc, Chord.

 

Pi: Irrational Math Constant

Pi Day.
Saturday, March 14, 2009 = 3.14
It's time to get irrational. Tomorrow is Pi Day, when mathematicians will gather to celebrate the mystery of science's most famous strange number.

 

Go Geometry Education Index

Go Geometry Education Index

 

Triangle, Circle, Radius, Perpendicular

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

 

Tangent Circles, Radius

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

 

Tangent Circles

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

 

Circula Sector, Semicircles, Circle

Proposed Problem 285.
Circular Sector 90 degrees, Semicircles, Circle, Tangent, Radius.

 

Elearning 284: Circular Sector

Proposed Problem 284.
Circular Sector 90 degrees, Semicircles, Tangent, Radius.

 

Circular Sector

Proposed Problem 283.
Circular Sector 90 degrees, Semicircle, Circle inscribed, Radius.

 

Elearning 279: Tangent circles

Proposed Problem 279.
Tangent Circles, Common External Tangent, Chords, Inradius.

 

Tangent Circles, Common External Tangent, Chord

Proposed Problem 278.
Tangent Circles, Common External Tangent, Chord.

 

Tangent Circles

Proposed Problem 277.
Tangent Circles, Common External Tangent.

 

Square, Circle, 90 Arc

Proposed Problem 276.
Square, 90 degree Arcs, Circle, Radius.

 

Sagitta Problem

Proposed Problem 275.
Right Triangle, Circumcircle, Sagitta, Inradius.

 

Cube of the external common tangent

Proposed Problem 271.
Tangent Circles, the Cube of the Common external tangent.

 

Tangent circles, common tangent 

Proposed Problem 270.
Tangent Circles, Common external tangent, Fractional exponents.

 

Regular hexagon inscribed in a circle

Proposed Problem 262.
Regular Hexagon inscribed in a circle, sum of distances.

 

Regular pentagon inscribed

Proposed Problem 261.
Regular Pentagon inscribed in a circle, sum of distances.

 

Problem 257: Equilateral triangle, circumcircle

Proposed Problem 257.
Equilateral Triangle, Circumcircle, Point, Vertices, Side, Distances, Squares.

 

Elearning 256: Equilateral triangle

Proposed Problem 256.
Equilateral Triangle, Circumcircle, Point, Vertices, Distances.

 

Elearning: Napoleon Theorem III

Proposed Problem 248.
Napoleon's Theorem III. Inner and outer Napoleon triangles, Area.

 

Napoleon Theorem II

Proposed Problem 247.
Napoleon's Theorem II. Internal Equilateral triangles. Inner Napoleon triangle.

 

Napoleon Theorem I

Proposed Problem 246.
Napoleon's Theorem I. External Equilateral triangles. Outer Napoleon triangle.

 

Right triangle, inradius, altitude, arithmetic mean

Proposed Problem 220. Right Triangle, Altitude, Angle Bisector, Distance, Arithmetic Mean.  

 

Elearn 215: Cyclic quadrilateral

Proposed Problem 215. Quadrilateral, Angle Bisectors, and Cyclic Quadrilateral.

 

Elearn 213 Triangle Inradius, Common Tangents

Proposed Problem 213. Triangle, Incircle, Inradius, Semicircles, Common Tangents.  

 

Archimedes Arbelos and Square

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

 

Archimedes Arbelos and Square

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

 

Elearn 209

Proposed Problem 209. Triangle, Incircles, Inradius.  

 

Geometry: Triangle Area, inradius, exradii

Proposed Problem 195. Area of a Triangle, Inradius, Exradii.  

 

Elearn 208: Triangle, Angles

Proposed Problem 208. Triangle, Excircles, Angles, 360 degrees.  

 

Problem 207: Hypotenuse

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

 

Elearn 206

Proposed Problem 206. Area of a Right Triangle, Inradius, andExradius relative to the hypotenuse.  

 

Right triangle area

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

 

Elearn 204

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

 

Problem 203

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

 

Problem 202 Right triangle, excircles

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

 

Elearn 201

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

 

Elearning 200: Right Triangle

Proposed Problem 200. RightTriangle, Incircle, Excircles, Points of Tangency, Inradius.

 

Elearning 197: Triangle Area

Proposed Problem 197. Area of a Triangle, Side, Inradius, and Exradius.

 

Elearning 196: Inradius, Exradii

Proposed Problem 196. Triangle, Inradius and Exradii Formula.

 

Geometry: Triangle Area, inradius, exradii

Proposed Problem 195. Area of a Triangle, Inradius, Exradii.

 

Area of a triangle, semiperimeter, exradius

Proposed Problem 194. Area of a Triangle, Semiperimeter, Exradius.

 

Area of a triangle, semiperimeter, inradius

Proposed Problem 193. Area of a Triangle, Semiperimeter, Inradius.

 

Geometry Problem Circle, Diameter, and Chord

Proposed Problem 192. Circle, Diameter, Chord, Perpendicular, Triangle, Area.

 

Elearning 190 tangent circles, chord, perpendicular

Proposed Problem 190. Tangent circles, Tangent chord, Perpendicular, Distance.

 

Elearning 187: Right triangle, incircles, angles

Proposed Problem 187. Right Triangle, Altitude, Incenters, Circles, Angles.

 

Elearning 186: Right triangle, circles

Proposed Problem 186. Right Triangle, Altitude, Incenters, Circles.

 

Overlapping circles, Angle

Proposed Problem 182. Overlapping Circles, Find an angle.

 

Circular Sector 90 degree

Proposed Problem 181. Circular Sector of 90 degrees, find an angle.

 

Circle tangent externally, trapezoid, area

Proposed Problem 180. Circles Tangent Externally, Common External Tangents, Areas.

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