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![Go Geometry Education](../geometry/h_online_math_geometry_910.jpg)
Geometry: Circles, Chord, Diameter,
Semicircle, Arc, Tangent, Theorems and Problems
Circular beauty,
Theorems and problems arise,
Inscribed in the arc.
Geometry's dance unfolds,
Circles reveal their secrets.
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![Circles](../geometry/circles_index_17.jpg) |
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![Circle, visual summary](circle-visual-summary-1.jpg) |
Circle, visual summary. |
![Geometry Problem 1569](../school-college/6/p1569-isosceles-triangle-square-relationship-diameter-incircle-tutoring-1.jpg) |
Geometry Problem 1569:
Prove a Relationship: EG Equals the Incircle Diameter in Triangle ABC with Square CDEF. |
![Geometry Problem 1567](../school-college/6/p1567-bases-isosceles-trapezoid-circumscribed-circle-chord-distance-measurement-tangent-tutoring-1.jpg) |
Geometry Problem 1567:
Finding Tangent Distances in a Circumscribed Isosceles Trapezoid. |
![Geometry Problem 1561](../school-college/6/p1561-highschool-college-geometry-triangle-ABC-circumcenter-orthocenter-angle-parallel-problem-1.jpg) |
Geometry Problem 1561:
Triangle ABC, Circumcenter O, Orthocenter H, Parallel Line, and Angle
Secrets. |
![Geometry Problem 1556](../school-college/6/p1556-right-triangle-inscribed-circle-properties-and-length-of-FG-problem-solving-1.jpg) |
Geometry Problem 1556: Right Triangle ABC and Inscribed Circle. The problem involves circle, chords, tangent, perpendicular lines, and congruence |
![Geometry Problem 1555](../school-college/6/p1555-circle-chord-intersect-geometry-problem-solving-1.jpg) |
Geometry Problem 1555: Find Length of DE. The problem involves chords, tangent, circles, and intersections of line |
![Geometry Problem 1553](../school-college/6/p1553-triangle-abc-angle-bisectors-calculate-circumradius-1.jpg) |
Geometry Problem 1553: Solving for OC in Triangle ABC with Unique Angle Bisectors |
![Geometry Problem 1551](../school-college/6/p1551-angle-relations-cyclic-quadrilateral-abcd-circumcircles-triangles-1.jpg) |
Geometry Problem 1551: Unraveling Angle Relations in Cyclic Quadrilaterals: Solving for Angle GEJ: A High School Level Investigation |
![Geometry Problem 1549](../school-college/6/p1549-triangle-incircle-tangent-inscribed-circle-angle-calculation-1.jpg) |
Geometry Problem 1549: Unraveling the Geometric Mystery: Calculating Angle
BGE with the Incircle and Tangent in Triangle ABC |
![Geometry Problem 1547](../school-college/6/p1547_circle_radius_4_tangents_AB_AC_points_of_tangency_distance_calculation_from_B_to_AC-1.jpg) |
Geometry Problem 1547: Tangents' Dance: Exploring B-to-AC Distance in a
Circle's Grasp |
![Geometry Problem 1546](../school-college/6/p1546-inscribed-circle-given-sides-calculate-contact-triangle-area-1.jpg) |
Geometry Problem 1546: Discover the Hidden Geometry: Calculate Area of Contact Triangle DEF in Triangle ABC |
![Geometry Problem 1545](../school-college/6/p1545-geometric-puzzle-area-of-triangle-inscribed-circle-excenters-incircle-1.jpg) |
Geometry Problem 1545: Unlock the Geometric Mystery: Calculate the Area of Triangle ABC with Inscribed Circle and Excenters! |
![Geometry Problem 1544](../school-college/6/p1544-calculate-area-triangle-with-given-arc-semicircle-intersection-geometry-resource-1.jpg) |
Geometry Problem 1544: Challenge: Calculate the Area of a Triangle with Given Arc and Semicircle Intersections. |
![Geometry Problem 1543](../school-college/6/p1543-square-circle-arc-mindpoint-quadrilateral-area-school-college-1.jpg) |
Geometry Problem 1543: Calculating the Area of Quadrilateral ABED in a Square with a Side Length of 20 and an Intersecting Arc. |
![Geometry Problem 1542](../school-college/6/p1542-right-triangle-square-circumscribed-inscribed-circle-hypotenuse-segment-length-angle-bisector-1.jpg) |
Geometry Problem 1542: Unraveling a Geometric Puzzle with a Circumscribed
Right Triangle and Square to the Same Circle for School and College. |
![Geometry Problem 1541](../school-college/6/p1541-inscribed-cyclic-quadrilateral-diameter-circumcircle-harmonic-quaternary-menelaus-ceva-1.jpg) |
Geometry Problem 1541 Challenge: Unveiling BG Length in an Inscribed Quadrilateral with Harmonic Quaternary Insight for Academic Pursuit. |
![Geometry Problem 1540](../school-college/6/p1540-length-chord-circle-intersection-diameter-similarity-math-class-1.jpg) |
Geometry Problem 1540: Solving for the Length of Chord in a Circle: Analyzing Intersections and Given Values for Academic Pursuits. |
![Geometry Problem 1539](../school-college/6/p1539-isosceles-triangle-tangent-circle-altitude-right-angle-midpoint-measurement-math-class-1.jpg) |
Geometry Problem 1539 Demystified: Unraveling the Lengths in an Isosceles Triangle with Altitude and Tangent Secrets!. |
![Geometry Problem 1535](../school-college/6/p1535-inscribed-circle-in-square-find-angle-crack-the-code-thematic-poem-1.jpg) |
Geometry
Problem 1535: Crack the Code: Inscribed Circle in Square - Angle Challenge! Solve the Mystery. |
![Geometry Problem 1534](../school-college/6/p1534-tangent-circle-common-external-tangent-angle-brainteaser-1.jpg) |
Geometry
Problem 1534: High School Brainteaser: Tangent Circles, Common External Tangent, and Angle Conundrums!.
. |
![Geometry Problem 1533](../school-college/6/p1533-square-exterior-right-triangle-find-angle-cyclic-quadrilateral-1.jpg) |
Geometry
Problem 1533: Discovering Relationships between Angles and Lines in an Exterior Right Triangle of a Square - A High School Challenge.
. |
![Geometry Problem 1532](../school-college/6/p1532-square-inner-semicircle-tangent-find-angle-chord-1.jpg) |
Geometry
Problem 1532: Crack the Code of Geometry Problem 1532: How to Find the Angle in a Square with a Tangent Semicircle! - A High School Challenge.
. |
![Geometry Problem 1531](../school-college/6/p1531-circle-diameter-chord-angle-pythagorean-theorem-length-30-degree-1.jpg) |
Geometry
Problem 1531: Discover How to Calculate the Length of a Chord in a Circle with Diameter Intersection and an Angle between the Diameter and Chord - A High School Challenge.
. |
![Geometry Problem 1528](../school-college/6/p1528-90-degree-circular-sector-inscribed-circle-tangent-find-angle-radius-1.jpg) |
Geometry
Problem 1528: Cracking the Circle Code: Unveiling the Tangent and Angle of an Inscribed Circle within a 90-Degree Circular Sector.
. |
![Geometry Problem 1527](../school-college/6/p1527-two-intersecting-circles-angle-between-chords-1.jpg) |
Geometry
Problem 1527: Discovering the Hidden Angle: Solving the Puzzle of Two Intersecting Circles.
. |
![Geometry Problem 1521](../school-college/6/p1521-triangle-find-angle-side-two-diameter-circle-1.jpg) |
Geometry
Problem 1521 and a Thematic Poem.
Unlock the Secret to Finding the Measure of an Angle in a Triangle with Two Sides as Diameters of Circles. |
![Geometry Problem 1503](../school-college/6/p1503-triangle-transversal-line-tangent-incircle-distance-vertices-perpendicular-congruence-1.jpg) |
Geometry
Problem 1503.
Triangle, Incircle, Tangent, Congruence, Perpendicular. |
![Geometry Problem 1502](../school-college/6/p1502-rigth-triangle-incircles-measurement-inradius-square-quadrilateral-1.jpg) |
Geometry
Problem 1502.
Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral. |
![Geometry Problem 1500](../school-college/5/p1500-circle-tangent-secant-perpendicular-parallel-similarity-cyclic-quadrilateral-1.jpg) |
Geometry
Problem 1500.
Circle, Perpendicular, Tangent, Secant, Cyclic Quadrilateral, Parallel, Similarity, Measurement. |
![Geometry Problem 1498](../school-college/5/p1498-intersecting-circles-diameter-common-chord-secant-cyclic-quadrilateral-concyclic-collinear-points-1.jpg) |
Geometry
Problem 1498.
Intersecting Circles, Diameter, Common Chord, Secant, Cyclic Quadrilateral, Concurrent Lines, Concyclic and Collinear Points. |
![Geometry Problem 1495](../school-college/5/p1495-circle-parallel-chords-30-degree-square-radius-1.jpg) |
Geometry
Problem 1495.
Circle, Parallel Chords, 30 Degree Angle, Radius Squared. |
![Geometry Problem 1492](../school-college/5/p1492-right-triangle-altitude-incenter-angle-measurement-1.gif) |
Geometry
Problem 1492.
Right Triangle, Altitude, Incenters, Angle, Measurement. |
![Geometry Problem 1491](../school-college/5/p1491-cyclic-quadrilateral-diagonal-incircle-angle-measurement-1.gif) |
Geometry
Problem 1491.
Cyclic Quadrilateral, Diagonal, Incircle, Angle, Measurement. |
![Geometry Problem 1486](../school-college/5/p1486-right-tringle-altitude-incircle-tangent-measurement-1.jpg) |
Geometry
Problem 1486.
Right Triangle, Altitude, Incircle, Tangent, Measurement. |
![Geometry Problem 1485](../school-college/5/p1485-triangle-altitude-orthocenter-circle-tangent-measurement-1.jpg) |
Geometry
Problem 1485.
Triangle, Orthocenter, Altitude, Circle, Diameter, Tangent, Measurement. |
![Geometry Problem 1484](../school-college/5/p1484-right-triangle-three-incircles-inradius-altitude-measurement-1.jpg) |
Geometry
Problem 1484.
Right Triangle, Altitude, Incircles, Inradius, Measurement. |
![Dynamic Geometry 1483](../school-college/5/p1483-isosceles-right-triangle-excenter-perpendicular-measurement-1.jpg) |
Geometry
Problem 1483.
Isosceles Right Triangle, Excenter, Perpendicular, Measurement. |
![Dynamic Geometry 1481](../school-college/5/p1481-5-circumscribed-quadrilateral-tangent-pitot-ipad-1.jpg) |
Dynamic Geometry 1481.
Five Tangential or Circumscribed Quadrilaterals, Pitot Theorem, Congruence, Step-by-step Illustration. GeoGebra,
iPad. |
![Dynamic Geometry 1480](../school-college/5/p1480-cyclic-polygon-triangulation-sum-inradii-invariant-geogebra-ipad-1.jpg) |
Dynamic Geometry 1480.
Japanese Theorem for Cyclic Polygon, Sangaku, Triangulation, Non-intersecting Diagonals, Sum of Inradii, Invariant, Step-by-step Illustration. GeoGebra,
iPad. |
![Dynamic Geometry 1479](../school-college/5/p1479-triangle-circumcircle-angle-perpendicular-bisector-parallel-concyclic-geogebra-ipad-1.jpg) |
Dynamic Geometry 1479.
Triangle, Circumcircle, Angle Bisector,
Perpendicular Bisector, Chord, Concyclic Points, Parallel Lines,, Step-by-step Illustration. GeoGebra,
iPad. |
![Dynamic Geometry 1478](../school-college/5/p1478-reuschle-terquem-theorem-3-cevians-circumcircle-concurrent-geogebra-ipad-1.jpg) |
Dynamic Geometry 1478.
Cyclocevian, Reuschle-Terquem Theorem, Concurrent Cevians, Triangle, Circumcircle, Secant line, Step-by-step Illustration. GeoGebra,
iPad. |
![Dynamic Geometry 1477](../school-college/5/p1477-miquel-pentagram-theorem-circumcircle-5-concyclic-geogebra-ipad-1.jpg) |
Dynamic Geometry 1477.
Miquel's Pentagram Theorem, Pentagon, Triangle, Circumcircles, Concyclic Points, Step-by-step Illustration. GeoGebra,
iPad. |
![Dynamic Geometry 1475](../school-college/5/p1475-clifford-intersecting-circles-theorem-geogebra-1.jpg) |
Dynamic Geometry 1475.
Clifford Intersecting Circles Theorem, Step-by-step Illustration, GeoGebra, iPad. |
![Dynamic Geometry 1474](../school-college/5/p1474-butterfly-theorem-circle-chord-midpoint-geogebra-ipad-1.jpg) |
Dynamic Geometry 1474.
Butterfly Theorem, Circle, Chords, Midpoints, Step-by-step Illustration. |
![Dynamic Geometry 1473](../school-college/5/p1473-kosnita-theorem-triangle-circumcenter-concurrent-geogebra-1.jpg) |
Dynamic Geometry 1473.
Kosnita's Theorem, Triangle, Four Circumcenters, Concurrent Line, Step-by-step Illustration. |
![Dynamic Geometry 1472](../school-college/5/p1472-cyclic-quadrilateral-perpendicular-diagonal-rectangle-geogebra-1.jpg) |
Dynamic Geometry 1472.
Cyclic Quadrilateral, Perpendicular Diagonals, Rectangle, Step-by-step Illustration. |
![Dynamic Geometry 1471](../school-college/5/p1471-equilateral-triangle-incenter-concurrent-geogebra-ipad-1.jpg) |
Dynamic Geometry 1471.
Equilateral Triangle, Inside/Outside Point, Incenters, Tangency Points, Concurrent Lines, Step-by-step Illustration. |
![Dynamic Geometry 1470](../school-college/5/p1470-tangential-quadrilateral-circle-rhombus-geogebra-1.jpg) |
Dynamic Geometry 1470.
Tangential Quadrilateral, Incircles, Tangent,
Parallel, Rhombus, Step-by-step Illustration. |
![Dynamic Geometry 1469](../school-college/5/p1469-triangle-inradius-circumradius-midpoint-geogebra-1.jpg) |
Dynamic Geometry 1469.
Triangle, Circumradius, Inradius, Midpoints, Arcs, Sum of Distances, Step-by-step Illustration. |
![Dynamic Geometry 1468](../school-college/5/p1468-steiner-theorem-triangle-inradius-exradius-circumradius-ipad-geogebra-1.jpg) |
Dynamic Geometry 1468.
Steiner's Theorem, Triangle, Circumradius, Inradius, Sum of Exradii, Step-by-step Illustration. |
![Geometry Problem 1466](../school-college/5/p1466-tangential-quadrilateral-newton-line-incenter-geogebra-ipad-1.jpg) |
Geometry Problem 1466.
Tangential Quadrilateral, Newton Line, Incenter, Midpoint, Diagonal, Collinear Points.
Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1465](../school-college/5/p1465-tangential-quadrilateral-incenter-equal-sum-area-1.jpg) |
Geometry Problem 1465.
Tangential Quadrilateral, Incenter, Inscribed Circle, Equal Sum of Areas.
Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1458](../school-college/5/p1458-incircle-excircle-triangle-area-1.jpg) |
Geometry
Problem 1458.
Triangle, Incircles, Excircle, Area, Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1457](../school-college/5/p1457-orthocenter-triangle-similarity-circles-radii-1.jpg) |
Geometry
Problem 1457.
Altitudes, Circles, Similarity,
Product of the Inradii Lengths,
iPad. Step-by-step illustration using GeoGebra. |
![Geometry Problem 1456](../school-college/5/p1456-orthocenter-triangle-similarity-circles-geogebra-ipad-1.jpg) |
Geometry
Problem 1456.
Altitudes, Orthic Triangle, Circumcircle, Parallel lines, Similarity, Area,
iPad. Step-by-step illustration using GeoGebra. |
![Geometry Problem 1455](../school-college/5/p1455-nagel-point-incircle-congruence-geogebra-ipad-1.jpg) |
Geometry
Problem 1455.
Nagel Point, Excircles, Incircle, Congruent Segments,
iPad. Step-by-step illustration using GeoGebra. |
![Geometry Problem 1454](../school-college/5/p1454-cyclic-quadrilateral-altitude-geogebra-dynamic-ipad-1.jpg) |
Geometry
Problem 1454.
Intersecting Circles, Perpendicular Lines, Cyclic Quadrilateral, Collinear
Points,
iPad. Step-by-step illustration using GeoGebra. |
![Geometry Problem 1453](../school-college/5/p1453-circle-diameter-cyclic-quadrilateral-concyclic-1.jpg) |
Geometry
Problem 1453.
Two Semicircles, Cyclic Quadrilateral, Concyclic Points,
iPad. |
![Dynamic Geometry 1452](../school-college/5/p1452-cyclic-quadrilateral-sangaku-japanese-theorem-geogebra-1.jpg) |
Dynamic Geometry 1452.
Japanese Theorem, Sangaku,
Cyclic Quadrilateral, Incenter, Rectangle, Inradius. Step-by-step
illustration using GeoGebra. |
![Dynamic Geometry 1449](../school-college/5/p1449-salmon-line-geogebra-dynamic-geometry-igs-1.jpg) |
Dynamic Geometry 1449.
Salmon Line. Step-by-step
illustration using GeoGebra. |
![Dynamic Geometry 1448](../school-college/5/p1448-simson-line-geogebra-dynamic-1.jpg) |
Dynamic Geometry 1448.
Simson Line. Step-by-step
illustration using GeoGebra. |
![Problem 1446](../school-college/5/p1446-lemoine-line-tangent-geogebra-dynamic-1.jpg) |
Dynamic Geometry Problem 1446.
Lemoine Line, triangle, circumcircle, tangent, collinear points. Step-by-step animation using GeoGebra. |
![Problem 1443](../school-college/5/p1443-triangle-area-incenter-90-degree-circumcenter-1.jpg) |
Geometry Problem 1443.
Triangle Area, Incenter, Circumcenter, 90 Degree, Perpendicular, One-third
Area, Art, Poster, Typography, iPad Apps. |
![Problem 1442](../school-college/5/p1442-triangle-altitude-angle-measurement-1.jpg) |
Geometry Problem 1442.
Triangle, Altitudes, Orthocenter, Perpendicular, Congruent Angles,
Similarity, Measurement, Art, Poster, Typography, iPad Apps. |
![Problem 1441](../school-college/5/p1441-intersecting-circles-perpendicular-bisector-parallel-1.jpg) |
Geometry Problem 1441.
Intersecting Circles, Perpendicular Bisector,
Parallel Lines, Art, Poster, Typography, iPad Apps. |
![Problem 1440](../school-college/5/p1440-intersecting-circles-perpendicular-bisector-collinearity-1.jpg) |
Geometry Problem 1440.
Intersecting Circles, Perpendicular Bisector, Collinear Points, Art, Poster, Typography, iPad Apps. |
![Problem 1439](../school-college/5/p1439-rectangle-diagonal-perpendicular-circle-ipad-apps-1.jpg) |
Geometry Problem 1439.
Rectangle, Diagonal, Perpendicular, Circles, Areas, Art, Poster, Typography, iPad Apps. |
![Problem 1438](../school-college/5/p1438-triangle-incenter-circuncenter-centroid-parallel-90-degree-ipad-apps-1.jpg) |
Geometry Problem 1438.
Triangle, Incenter, Centroid, Circumcenter, Parallel, 90 Degree, Art, Poster, Typography, iPad Apps. |
![Problem 1437](../school-college/5/p1437-circle-intersecting-tangent-equal-product-ipad-sw-typography-1.jpg) |
Geometry Problem 1437.
Intersecting Circles, Common Tangent, Equal Product, Art, Poster, Typography, iPad Apps. |
![Problem 1436](../school-college/5/p1436-right-triangle-altitude-excircle-geometric-mean-1.jpg) |
Geometry Problem 1436.
Right Triangle, Altitude, Excircles, Excenters, Geometric Mean,
Measurement, Art,
Poster, Typography, iPad Apps. |
![Problem 1435](../school-college/5/p1435-circle-diameter-tangent-parallelogram-1.jpg) |
Geometry Problem 1435.
Circle, Diameter, Inscribed Circles, Circular Sector, Parallelogram, Parallel Lines, Tangency Points,
Poster, Typography, iPad Apps. |
![Problem 1434](../school-college/5/p1434-equilateral-triangle-tangent-circles-line-1.jpg) |
Geometry Problem 1434.
Equilateral Triangle, Tangent Circles, Tangent Lines, Arithmetic Mean, Measurement,
Poster, Typography, iPad Apps. |
![Problem 1433](../school-college/5/p1433-tangent-circles-line-midpoint-1.jpg) |
Geometry Problem 1433.
Tangent Circles, Diameter, Perpendicular, Midpoint, Measurement,
Poster Typography, iPad Apps. |
![Problem 1432](../school-college/5/p1432-tangent-circles-proportion-similarity-ipad-apps-1.jpg) |
Geometry Problem 1432.
Tangent Circles, Secant, Tangent Lines, Proportionality, Similarity,
Poster Typography, iPad. |
![Problem 1431](../school-college/5/p1431-apollonius-llc-triangle-circumcircle-tangent-congruent-angle-1.jpg) |
Geometry Problem 1431.
Triangle, Circumcircle, Parallel Chord, Apollonius, LLC, Congruent Angles, Poster, Conformal Mapping. |
![Problem 1428](../school-college/5/p1428-circle-tangent-triangle-square-area-1.jpg) |
Geometry Problem 1428. Intersecting Circles, Tangent Line, Triangle, Square, Area. |
![Problem 1427](../school-college/5/p1427-square-circle-tangent-radius-1.jpg) |
Geometry Problem 1427. Circle inscribed in a square, Arc, Tangent Line, Tangent Circles, Radii. |
![Problem 1423](../school-college/5/p1423-triangle-equal-angle-similarity-area-1.jpg) |
Geometry Problem 1423. Triangle, Area, Three Equal Angles, Circumradius, Measurement. |
![Problem 1421](../school-college/5/p1421-right-triangle-incircle-excircle-measurement-1.jpg) |
Geometry Problem 1421. Right Triangle, Incircle, Excircle, Tangent Lines, Measurement. |
![Problem 1420](../school-college/5/p1420-circle-area-regular-hexagon-tangent-1.jpg) |
Geometry Problem 1420. Regular Hexagon, Inscribed Circle, Area, Tangent
Circles, Tangent Line. |
![Problem 1419](../school-college/5/p1419-triangle-incenter-inradius-circumcircle-distance-sketch-ipad-1.jpg) |
Geometry Problem 1419. Triangle, Altitude, Circumcircle, Incenter, Inradius, Perpendicular, Angle Bisector, Midpoint. |
![Problem 1418](../school-college/5/p1418-triangle-incenter-parallel-angle-circle-tangent-1.jpg) |
Geometry Problem 1418. Triangle, Circumcircle, Incenter, Side, Sum of Segments, Tangent, Parallel, Angle. |
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Last updated:Jul 26, 2024
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