Online Geometry

Triangle Centers & Inscribed Figures

The Incenter: The Heart of Triangle Harmony

Journey to the internal equilibrium of the triangle. The incenter, defined by the perfect convergence of angle bisectors, serves as the center of the incircle—a sanctuary of tangency that touches every side with mathematical grace. Explore the theorems of the inradius and the properties that govern this fundamental point of geometric balance.

Incenter, Incircle, Index

  

Euclid's Elements 23 definitions

Euclid's Elements Book I, 23 Definitions. One-page visual illustration.
Euclid's Elements Book. Index

Triangle Centers.

Distance between triangle centers

Distances between Triangle Centers Index.

Dynamic Geometry: Incenter and Incircle of a Triangle. HTML5 Animation for Tablets (iPad, Nexus..)

GeoGebra, Dynamic Geometry: Incenter and Incircle of a Triangle.
HTML5 Animation for Tablets (iPad, Nexus..)

Gergonne Point Theorems and Problems Index

Gergonne Points Index
Triangle Center

Nagel Point Theorems and Problems Index

Nagel Points Index
Triangle Center

Right Triangle, Altitude, Incircle Index

Right triangle, Altitude, Incircle
Right Triangle, Altitude to the Hypotenuse, Incircle, Incenter, Inradius, Angle Bisector, Theorems and Problems, Index.

Sawayama -Thebault's theorem
Incenter, Incircle, Circumcircle.

Distance between the incenter to the centroid

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

Geometry Problem 1616

Geometry Problem 1616: Incenter and Segment Ratios in a Contact Triangle

Geometry Problem 1608

Geometry Problem 1608: Lunes of Hippocrates Area from Incenter Semicircles

Geometry Problem 1590

Geometry Problem 1590: Midpoints, Incircles, Areas, and the Art of Balance.

Geometry Problem 1585

Geometry Problem 1585: Discover how a triangle's incircle reveals the secret of harmonious convergence

Geometry Problem 1584

Geometry Problem 1584: Proving a Ratio Involving Incircle and Angle Bisector
Unveiling a Potentially Unexplored Geometric Relationshp.

Geometry Problem 1569

Geometry Problem 1569: Prove a Relationship: EG Equals the Incircle Diameter in Triangle ABC with Square CDEF.

Geometry Problem 1568

Geometry Problem 1568: Concyclicity of Points B, D , H, J in a Triangle ABC with an Incircle and a Tangent Circle.

Geometry Problem 1556

Geometry Problem 1556: Right Triangle ABC and Inscribed Circle. The problem involves circle, chords, tangent, perpendicular lines, and congruence

Geometry Problem 1549

Geometry Problem 1549: Unraveling the Geometric Mystery: Calculating Angle BGE with the Incircle and Tangent in Triangle ABC

Geometry Problem 1546

Geometry Problem 1546: Discover the Hidden Geometry: Calculate Area of Contact Triangle DEF in Triangle ABC

Geometry Problem 1545

Geometry Problem 1545: Unlock the Geometric Mystery: Calculate the Area of Triangle ABC with Inscribed Circle and Excenters!

Geometry Problem 1542

Geometry Problem 1542: Unraveling a Geometric Puzzle with a Circumscribed Right Triangle and Square to the Same Circle for School and College.

Geometry Problem 1503

Geometry Problem 1503.
Triangle, Incircle, Tangent, Congruence, Perpendicular.

Geometry Problem 1502

Geometry Problem 1502.
Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral.

Geometry Problem 1492

Geometry Problem 1492.
Right Triangle, Altitude, Incenters, Angle, Measurement.

Geometry Problem 1491

Geometry Problem 1491.
Cyclic Quadrilateral, Diagonal, Incircle, Angle, Measurement.

Geometry Problem 1490

Geometry Problem 1490.
Triangle, Cevian, Incircle, Tangent, Measurement.

Geometry Problem 1489

Geometry Problem 1489.
Right Triangle, Angle Bisectors, Perpendicular, Measurement.

Geometry Problem 1488

Geometry Problem 1488.
Right Triangle, Altitude, Incircle, Tangent, Measurement.

Geometry Problem 1487

Geometry Problem 1487.
Right Triangle, Altitude, Incircle, Tangent, Measurement.

Geometry Problem 1486

Geometry Problem 1486.
Right Triangle, Altitude, Incircle, Tangent, Measurement.

Geometry Problem 1484

Geometry Problem 1484.
Right Triangle, Altitude, Incircles, Inradius, Measurement.

Dynamic Geometry 1481

Dynamic Geometry 1481.
Five Tangential or Circumscribed Quadrilaterals, Pitot Theorem, Congruence, Step-by-step Illustration. GeoGebra, iPad.

Dynamic Geometry 1480

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 1471

Dynamic Geometry 1471.
Equilateral Triangle, Inside/Outside Point, Incenters, Tangency Points, Concurrent Lines, Step-by-step Illustration.

Dynamic Geometry 1470

Dynamic Geometry 1470.
Tangential Quadrilateral, Incircles, Tangent, Parallel, Rhombus, Step-by-step Illustration.

Dynamic Geometry 1469

Dynamic Geometry 1469.
Triangle, Circumradius, Inradius, Midpoints, Arcs, Sum of Distances, Step-by-step Illustration.

Dynamic Geometry 1468

Dynamic Geometry 1468.
Steiner's Theorem, Triangle, Circumradius, Inradius, Sum of Exradii, Step-by-step Illustration.

Geometry Problem 1466

Geometry Problem 1466.
Tangential Quadrilateral, Newton Line, Incenter, Midpoint, Diagonal, Collinear Points. Step-by-step Illustration using GeoGebra.

Geometry Problem 1465

Geometry Problem 1465.
Tangential Quadrilateral, Incenter, Inscribed Circle, Equal Sum of Areas. Step-by-step Illustration using GeoGebra.

Geometry Problem 1458

Geometry Problem 1458.
Triangle, Incircles, Excircle, Area, Step-by-step Illustration using GeoGebra.

Geometry Problem 1457

Geometry Problem 1457.
Altitudes, Circles, Similarity, Product of the Inradii Lengths, iPad. Step-by-step illustration using GeoGebra.

Geometry Problem 1455

Geometry Problem 1455.
Nagel Point, Excircles, Incircle, Congruent Segments, iPad. Step-by-step illustration using GeoGebra.

Dynamic Geometry 1452

Dynamic Geometry 1452.
Japanese Theorem, Sangaku, Cyclic Quadrilateral, Incenter, Rectangle, Inradius. Step-by-step illustration using GeoGebra.

Problem 1443

Geometry Problem 1443.
Triangle Area, Incenter, Circumcenter, 90 Degree, Perpendicular, One-third Area, Art, Poster, Typography, iPad Apps.

Problem 1438

Geometry Problem 1438.
Triangle, Incenter, Centroid, Circumcenter, Parallel, 90 Degree, Art, Poster, Typography, iPad Apps.

Problem 1421

Geometry Problem 1421.
Right Triangle, Incircle, Excircle, Tangent Lines, Measurement.

Problem 1419

Geometry Problem 1419.
Triangle, Altitude, Circumcircle, Incenter, Inradius, Perpendicular, Angle Bisector, Midpoint.

Problem 1418

Geometry Problem 1418.
Triangle, Circumcircle, Incenter, Side, Sum of Segments, Tangent, Parallel, Angle.

Problem 1417

Geometry Problem 1417.
Triangle, Cevian, Circumcircle, Incenter, Perpendicular, Angle Bisector, Tangent Lines, Tangent Circles.

Problem 1416

Geometry Problem 1416.
Right Triangle, Altitude, Incircle, Excircle, Tangency Points, 45 Degree Angle.

Problem 1415

Geometry Problem 1415.
Right Triangle, Altitude, Incircle, Excircle, Tangency Points, Isosceles Triangle.

Problem 1414

Geometry Problem 1414.
Right Triangle, Altitude, Incircle, Excircle, Tangency Points, Isosceles Triangle.

Problem 1413

Geometry Problem 1413.
Right Triangle, Incircle, Excircle, Tangency Points, Isosceles Right Triangle.

Problem 1412

Geometry Problem 1412.
Right Triangle, Incircle, Excircle, Tangency Points, 45 Degree Angle.

Problem 1411

Geometry Problem 1411.
Right Triangle, Incircle, Excircle, Tangency Points, Isosceles Right Triangle.

Problem 1410

Geometry Problem 1410.
Right Triangle, Incircle, Excircle, Tangency Points, 45 Degree Angle.

Problem 1409

Geometry Problem 1409.
Right Triangle, Incircle, Excircle, Collinear Tangency Points, Collinearity.

Problem 1408

Geometry Problem 1408.
Right Triangle, Incircle, Excircle, Incenter, Midpoint, Tangency Point, Collinearity.

Problem 1407

Geometry Problem 1407.
Right Triangle, Incircle, Excircle, Collinear Tangency Points, Collinearity.

Problem 1388 40-60-80 triangle

Geometry Problem 1388.
Triangle 40-60-80 degree, Incenter, Congruence.

Geometry problem 1350

Geometry Problem 1350.
Triangle with three Intersecting Circles, Incircle, Collinear Points.

Geometry problem 1349

Geometry Problem 1349.
Three Squares, 90 Degrees, Perpendicular Lines, Diagonal. Math Infographic.

Geometry problem 1348

Geometry Problem 1348.
Three Concentric Circles, Equilateral Triangle, Perpendicular Bisector, Congruence.

Geometry problem 1347

Geometry Problem 1347.
Triangle, Angle Bisector, Congruence, Circumcircle, Tangent, Equilateral Triangle.

Geometry problem 1346

Geometry Problem 1346.
Equilateral Triangle, Point on the Incircle, Altitude, Perpendicular, Sum of Squares, Distance.

Geometry problem 1345

Geometry Problem 1345.
Three Equilateral Triangles, Center, Area, Common Vertex.

Geometry problem 1344 Bottema Theorem. Elearning.

Geometry Problem 1344, Bottema's Theorem: Triangle and Squares with Interactive Geometry Software.

Geometry problem 1343

Problem 1343.
Triangle, Excircles, Circle, Tangent, Tangency Points, Chord, Perpendicular, 90 Degrees, Collinearity. Machu Picchu in the background.

Geometry problem 1342

Problem 1342.
Circle, Secant, Chord, Midpoint, Concyclic Points, Cyclic Quadrilateral.

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Last updated: Apr 18, 2026