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SYMMETRY & PROPORTION
The Geometry of Equity: Angle Bisectors and Their Theorems
The angle bisector is the architect of geometric balance. By tracing the path of points equidistant from two lines,
it reveals a profound symmetry that governs triangles, polygons, and circles alike.
Dive into our collection of theorems and problems to discover how this "line of harmony" unfolds the secrets of proportion and concurrency.
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Euclid's Elements
Book I, 23 Definitions. One-page visual illustration. Angles and more.
Euclid's Elements Book.
Index. |
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Book I, Proposition 9: Bisect a given angle.
Construction of angle bisector
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Incenter. Index.
Angle bisectors
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Euclid's Elements Book VI, Proposition 3: Angle Bisector Theorem |
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Isogonal Lines.
Theorems and Problems. |
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GeoGebra, Dynamic Geometry: Incenter and Incircle
of a Triangle.
HTML5 Animation for Tablets (iPad, Nexus..) |
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Animated
illustration of the celebrated Morley's angle trisector theorem |
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Morley's Triangle & Center: Trisection of angles. |
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Blanchet Theorem.
Altitude, Angle Bisector.
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Isogonic-Jacobi
Theorem: Using TracenPoche Dynamic
Geometry Software.
Isogonal conjugate line,
reflecting line on the angle bisector |
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Triangle: Incircle, Perpendicular, Angle Bisector.
Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Prove proposition. |
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Heron's Formula.
Key facts and
a purely geometric
step-by-step proof.
Geometry help: Key Facts: Angle bisector.
Preliminary propositions |
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Geometry Problem 1616: Incenter and Segment Ratios in a Contact Triangle |
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Geometry Problem 1611: Angle Calculation in Triangle ABC with Auxiliary Construction |
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Geometry Problem 1605: Triangle, Angle Bisector, and Circumcircle |
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Geometry Problem 1594: Intersecting Squares & Hidden Lengths � A STEM Perspective |
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Geometry Problem 1593: Find BF in Right Triangle ABC � A STEM Perspective |
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Geometry Problem 1591: Unravel Quadrilateral ABCD with Three Congruent Sides and Proportional Angles to Find Angle D. |
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Geometry Problem 1591: Unravel Quadrilateral ABCD with Three Congruent Sides and Proportional Angles to Find Angle D. |
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Geometry Problem 1581:
Prove the Angle Bisection in a Cyclic Quadrilateral. |
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Geometry Problem 1575:
Prove an Angle Bisector in a Triangle Involving an Altitude, Midpoint, and Excircle |
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Geometry Problem 1571:
Prove that the intermediate angle in a Pythagorean Triple 7-24-25 measures 74 degrees. |
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Geometry Problem 1564:
Find the Area of Quadrilateral BGDJ in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. |
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Geometry Problem 1563:
Perpendicularity in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. |
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Geometry Problem 1562:
Proof of Collinearity in a Right-angled Triangle involving the Altitude, Angle bisectors, and Midpoint. |
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Geometry Problem 1553: Solving for OC in Triangle ABC with Unique Angle Bisectors |
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Geometry Problem 1550: Solving for Segment BD: An Angle Puzzle in Right Triangle ABC: A High School Level Investigation |
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Geometry Problem 1549: Unraveling the Geometric Mystery: Calculating Angle
BGE with the Incircle and Tangent in Triangle ABC |
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Geometry Problem 1547: Tangents' Dance: Exploring B-to-AC Distance in a
Circle's Grasp |
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Geometry Problem 1542: Unraveling a Geometric Puzzle with a Circumscribed
Right Triangle and Square to the Same Circle for School and College. |
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Geometry
Problem 1526: Mastering Geometry Problem-Solving: Discover the Distance Between Two Sides in a Parallelogram Using Bisectors and Distance Measures.
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Geometry
Problem 1524 and a Thematic Poem.
Unlock the Mystery of Parallelograms: Discover the Length of Segment between the Intersecting Angle Bisectors. |
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Geometry
Problem 1518 and a Thematic Poem.
Boost Your Geometry Skills: Solve for the Number of Sides in an Equiangular Polygon with an Interior Point and Bisected Angle. |
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Geometry
Problem 1505.
Parallel Rays, Angles, and Bisector, Corresponding, Alternate, Conjugate. |
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Geometry
Problem 1502.
Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral. |
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Geometry
Problem 1491.
Cyclic Quadrilateral, Diagonal, Incircle, Angle, Measurement. |
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Geometry
Problem 1490.
Triangle, Cevian, Incircle, Tangent, Measurement. |
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Geometry
Problem 1489.
Right Triangle, Angle Bisectors, Perpendicular, Measurement. |
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Geometry
Problem 1488.
Right Triangle, Altitude, Incircle, Tangent, Measurement. |
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Geometry
Problem 1487.
Right Triangle, Altitude, Incircle, Tangent, Measurement. |
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Dynamic Geometry 1480.
Japanese Theorem for Cyclic Polygon, Sangaku, Triangulation, Non-intersecting Diagonals, Sum of Inradii, Invariant, Step-by-step Illustration. GeoGebra,
iPad. |
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Dynamic Geometry 1479.
Triangle, Circumcircle, Angle Bisector,
Perpendicular Bisector, Chord, Concyclic Points, Parallel Lines,, Step-by-step Illustration. GeoGebra,
iPad. |
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Geometry Problem 1419.
Triangle, Altitude, Circumcircle, Incenter, Inradius, Perpendicular, Angle Bisector, Midpoint. |
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Geometry Problem 1418.
Triangle, Circumcircle, Incenter, Side, Sum of Segments, Tangent, Parallel, Angle. |
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Geometry Problem 1417.
Triangle, Cevian, Circumcircle, Incenter, Perpendicular, Angle Bisector,
Tangent Lines, Tangent Circles. |
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Geometry Problem 1385.
Triangle, Orthocenter, Circle, Circumcircle, Angle Bisector, Midpoint, Collinear Points. |
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Geometry Problem 1384.
Triangle, Orthocenter, Circle, Circumcircle, Angle Bisector, 90 Degree. |
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Geometry Problem 1366.
Square, Angle Bisector, Perpendicular, Concyclic Points, Circle, Sketch, iPad Apps. |
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Geometry Problem 1364.
Square, Inscribed Circle, Area, Angle Bisector, Perpendicular, 45 Degrees, Sketch, iPad Apps. |
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Geometry Problem 1363.
Square, Angle Bisector, Perpendicular, 45 Degrees, Angle, Sketch, iPad Apps. |
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Geometry Problem 1347.
Triangle, Angle Bisector, Congruence, Circumcircle, Tangent, Equilateral Triangle. |
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Problem 1341.
Geometry Problem 1341: Isosceles Triangle, 80-20-80 Degrees, Circumcenter, Angle Bisector.
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Geometry Problem 1326.
Triangle, Cevian, Incenters, Sum of Angles, 270 Degrees. |
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Geometry Problem 1324.
Quadrilateral, Diagonal, 45 Degrees, Angle Bisector, Isosceles Triangle, Congruence. |
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Geometry Problem 1322.
Triangle, Angle Bisector, Circumcircle, Chord, Secant, Sum of two Angles. |
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Geometry Problem 1307.
Triangle, Incenter, Parallel line, Sides, Measurement. |
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Geometry Problem 1305.
Triangle, Circumcircle, Angle Bisector, Arc, Perpendicular, Area. |
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Geometry Problem 1304.
Triangle, 120 Degrees, Angle Bisector, Measurement. |
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Geometry Problem 1303.
Triangle, 60 Degrees, Angle Bisector, Measurement. |
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Geometry Problem 1298.
Arbelos, Semicircles, Diameters, Circle, Incircle, Tangent, Angle Bisector, Perpendicular, Midpoint. |
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Geometry Problem 1297.
Arbelos, Semicircles, Diameters, Circle, Incircle, Tangent, Angle Bisector. |
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Geometry Problem 1290.
Triangle, Internal Angle Bisector, Median, Parallel, Measurement, Metric Relations. |
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Geometry Problem 1274.
Isosceles Triangle, 80-20-80 Degrees, Area, Inradius, Circumradius, Angle Bisector, Metric Relations, Measurement. |
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Geometry Problem 1271.
Triangle, Incircle, Incenter, Excircle, Excenter, Escribed Circle, Tangency Points, Six Concyclic Points. |
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Geometry Problem 1270.
Triangle, Excircle, Excenter, Escribed Circle, Tangency Points, Perpendicular, 90 Degrees, Angle Bisector. |
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Geometry Problem 1269.
Triangle, Incircle, Incenter, Inscribed Circle, Tangency Points, Perpendicular, 90 Degrees, Angle Bisector. |
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Geometry Problem 1268.
Triangle, Exterior Angle Bisector, Circumcircle, Circle, Chord, Parallel, Concyclic Points, Cyclic Quadrilateral. |
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Geometry Problem 1266.
Triangle, Excircle, Circle, Tangency Points, Perpendicular, 90 Degrees, Angle Bisector. |
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Geometry Problem 1265.
Triangle, Incircle, Circle, Tangency Points, Perpendicular, 90 Degrees, Angle Bisector. |
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Geometry Problem 1264.
Triangle, Exterior Angle Bisector, Circumcircle, Circle, Perpendicular, 90 Degrees, Concurrent Lines. |
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Geometry Problem 1260.
Triangle, Internal Angle Bisector, Circumcircle, Circle, Perpendicular, 90 Degrees, Concurrent Lines. |
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Geometry Problem 1256.
Parallelogram, Angle Bisector, Measurement. |
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Geometry Problem 1249.
Cyclic Quadrilateral, Circle, Triangle, Circumcircle, Angle Bisector, Parallel Lines, Area. |
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Geometry Problem 1243.
Quadrilateral, Four Exterior Angle Bisectors, Semi-sum, Angles. Mobile Apps. |
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Geometry Problem 1242.
Quadrilateral, Four Interior and Exterior Angle Bisectors, Collinearity, Concurrent Lines. Mobile Apps. |
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Geometry Problem 1241.
Quadrilateral, Four Exterior Angle Bisectors, Cyclic Quadrilateral, Circle. Mobile Apps. |
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Geometry Problem 1231
Triangle, Orthocenter, Incenter, Circumcenter, Angle Bisector, Center, Circle. |
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Geometry Problem 1227
Triangle, Circumcircle, Angle Bisector, Concyclic Points, Cyclic Quadrilateral. |
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Geometry Problem 1225
Equilateral Triangle, Circumcircle, Angle Bisector, Congruence. |
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Geometry Problem 1217
Triangle, Circle, Excenter, Incenter, Angle Bisector, Cyclic Quadrilateral, Circumcircle, Tangent Line. |
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Geometry Problem 1214
Triangle, Three Angle Bisectors, Proportions. |
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Geometry Problem 1211
Right Triangle, Altitude, Angle Bisector, 45 Degrees. |
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