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

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

Circular Sector. Index.
Theorems and Problems. 

Concentric Circles. Index.
Theorems and Problems. 

Intersecting Circles Index.
Theorems and Problems.

Tangent Circles Index
 

Word Cloud of Circle.
Visual Summary.

Circumcenter. Index.

Incenter of a triangle. Index.

Excenter. Index.
Excircle, Exradius.

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

Mascheroni construction with compass alone. Index

Cyclic Quadrilateral.
 

Tangential or Circumscribed Quadrilateral
 

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

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

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

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: 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: On a given finite line to construct an equilateral triangle

Distance between the Incenter and the Centroid of a Triangle.
Formula in terms of the sides a,b,c.

Distances between Triangle Centers Index.

Tangential or Circumscribed Quadrilateral: Diagonals, Inradii Theorem.

Six Tangential or Circumscribed Quadrilaterals Theorem.

Tangential or Circumscribed Quadrilateral: Pitot Theorem.

Problem 493.
Right Triangle, Circumcircle, Perpendicular, Chord.

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

Problem 491.
Square, Right Triangle, Incircle, Inscribed Circle, Radius.

Problem 490.
Tangent Circles, Arc, Radius, Circular Sector, Area.

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

Problem 486.
Tangent circles, Chord, Arc, Midpoint, Collinearity, Congruence.

Problem 485.
Square, Side, Circles, Tangent, Radius, Measurement.

Problem 482.
Triangle, Circumcircle, Incenter, Excenter, Midpoint, Cyclic Points.

Problem 480.
Triangle, Circle, Center, Altitude, Angle, Circumcenter.

Problem 478.
Triangle, Sides, Circumradius, Circumcenter, Circumcircle.

Problem 476.
Triangle, Circumcenter, Circumcircle, Angles.

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

Problem 474.
Parallelogram, Diagonal, Circle, Vertex.

Proposed Problem 468.
Circle, Chord, Midpoint, Congruence, Angle.

Proposed Problem 467.
Concentric Circles, Chord, Tangent, Congruence, Measurement.

Proposed Problem 466.
Concentric Circles, Secant, Congruence, Measurement.

Proposed Problem 465.
Square, Arc, Angle, 90 Degrees, Measurement.

Proposed Problem 464.
Square, Arcs, Center, Angle, 90 Degrees, 120 Degrees.

Proposed Problem 463.
Three Concentric Circles, Secant, Tangent, Triangle, Area.

Proposed Problem 462.
Square, Arcs, 90 Degrees, Circle, Tangent, Radius, Measurement.

Proposed Problem 461.
Three Circles, Tangent, Right Angle, Center, Distance, Measurement.

Proposed Problem 460.
Circle, Tangent, Perpendicular, Radius, Distance, Measurement.

Proposed Problem 458.
Square, Semicircle, Circular Sector, Internal Common Tangent, Measurement.

Proposed Problem 456.
Three tangent circles, Center, Chord, Secant, Collinear.

Proposed Problem 455.
Rhombus, Inscribed Circle, Angle, Chord, 45 Degrees.

Proposed Problem 454.
Triangle, Incircle, Inradius, Tangent, Circle, Radius.

Proposed Problem 453.
Intersecting Circles, Tangent, Circular Segment, Congruence, Area.

Proposed Problem 452.
Triangle, Circle, Tangent, Proportional Segments.

Proposed Problem 449.
Congruent Tangent Circles, Diameter, Sagitta, Arc, Perpendicular, Chord.

Proposed Problem 448.
Tangent Circles, Diameter, Perpendicular, Chord.

Proposed Problem 447.
Complete Quadrilateral, Inradii, Exradii, Sides.

Proposed Problem 445.
Three Tangent Circles, Tangency Points, Circumradius.

Proposed Problem 444.
Internally Tangent circles, Secant line, Chords, Angles, Congruence.

Proposed Problem 443.
Overlapping, Intersecting Circles, Common Chord, Secant line, Tangent line, Angles.

Proposed Problem 439.
Isosceles triangle, Cevian, Incenter, Angles, Circle.

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

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

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

Proposed Problem 435.
Triangle, Incircle, Inradius, Congruent circles, tangent, Measurement.

Proposed Problem 426.
Triangle, Circumradius, Circumcenter, concurrent Cevians.

Proposed Problem 419.
Triangle, Cevian, Incircle, Excircle, Inradius, Exradius

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

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

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

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

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

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

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

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

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

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

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 377.
Triangle, Internal Angle Bisector, Altitude.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 332.
Cyclic quadrilateral, Ratio of diagonals, Similarity.

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

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

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

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

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

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

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

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

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

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

Proposed Problem 316.
Circular segments and Inscribed Squares.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Geometry Expressions.

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

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

Congruence. Index.

Similarity, Ratios, Proportions. Index.

Areas Index

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