Online Geometry: Incenter, Incircle of a triangle, Theorems and Problems.

Incenter of a triangle - Table of Content

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

Triangle Centers.

Triangle Incenter by Folding, Video .
Dynamic Geometry Software: Tabula by Numeracy Works.
Step-by-Step construction, Manipulation, and animation.

Distances between Triangle Centers Index.

Gergonne Points Index
Triangle Center

Nagel Points Index
Triangle Center

First Ajima-Malfatti Point
Animated illustration.

Second Ajima-Malfatti Point
Interactive illustration.

Feuerbach Points and Nine-Point Circle with interactive animation, manipulation, and step-by-step construction.
Incenter, Incircle, Excircles.

Problem 79: Triangle. Similarity, Altitudes, Orthocenter, Incenter, Incircles, Inradii.

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

Incenter, Excenter, Incircle, Excircle Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation.

Three Tangent Circles Theorem Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation.
Incenter of a triangle.

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

Geometry Problem 719.
Incenter, Intersecting Circles, Angle, Measurement.

Geometry Problem 713.
Tangent Circles, Diameter, Chord, Tangent, Incenter, Incircle, Triangle.

Geometry Problem 703.
Triangle, Incircle, Incenter, Tangency point, Diameter, Congruence.

Geometry Problem 684: Outer Nagel Point.
Triangle, Excircles, Incircle Tangency points, Concurrent lines, Semi-perimeter, Ceva theorem.

Geometry Problem 681: Gergonne Point.
Triangle, Incircle, Tangency points, Concurrent lines, Semiperimeter, Ceva theorem.

Geometry Problem 670.
Triangle, 60 Degrees, Orthocenter, Incenter, Circumcenter, Congruence.

Geometry Problem 669.
Triangle, Circumcircle, Incenter, Midpoint, Collinear points, Mind Map.

Geometry Problem 668.
Right triangle, Incircles, Parallel, Common Tangent, Mind Map.

Geometry Problem 664.
Triangle, 60 Degrees, Orthocenter, Incenter, Circumcenter, Angles, Triangle centers.

Geometry Problem 662.
Triangle, Incenter, Circumcenter, Orthocenter, Circle, Angle, Measurement.

Geometry Problem 657.
Triangle, Incircle, Tangency Points, Parallel Lines, Congruence.

Geometry Problem 634
Right Triangle, Altitude, Incenters, Areas.

Geometry Problem 626
Triangle, Distance from the Incenter to an Excenter.

Geometry Problem 624
Triangle, Incenter, Excenter, Incircle, Excircle, Similarity.

Geometry Problem 623
Triangle, Incenter, Excenter, Incircle, Excircle, Concyclic Points.

Geometry Problem 619
Triangle Area, 30 Degrees, Incircle, Incenter, Tangent.

Geometry Problem 618
Triangle Area, 60 Degrees, Incircle, Incenter, Tangent.

Geometry Problem 617
Right Triangle Area, Incircle, Incenter, Tangent, 90 Degrees.

Geometry Problem 594.
Triangle, Incenter, Incircle, Inradius, Tangency Point, Midpoint, Altitude, Angle, Half the Difference.

Geometry Problem 592.
Triangle, Incenter, Incircle, Tangency Point, Midpoints, Concurrent Lines, Congruence.

Geometry Problem 591.
Triangle, Incenter, Incircle, Inradius, Tangency Point, Midpoint, Altitude.

Geometry Problem 590.
Triangle, Incenter, Incircle, Tangency Point, Midpoint, Areas.

Geometry Problem 589.
Triangle, Incenter, Incircle, Tangency Points, Midpoint, Perpendicular, Distance.

Geometry Problem 588.
Triangle, Incenter, Incircle, Tangency Points, Midpoints, Distance, Sum.

Geometry Problem 587.
Triangle, Incenter, Incircle, Tangency Point, Midpoint, Distance, Half the Difference.

Problem 582.
Quadrilateral, Diagonals, Incenters, Metric Relations.

Problem 567.
Right triangle, Incircle, Excircle, Collinear tangency points.

Problem 564.
Complete Quadrilateral, Incenter, Excenter, Concyclic Points, Incircle, Excircle.

Problem 562.
Triangle, Cevian, Incenter, Perpendicular, Concyclic Points, Cyclic Quadrilateral.

Problem 561.
Triangle, Cevian, Incenter, Incircle, Perpendicular, Tangent.

Problem 560.
Triangle, Cevian, Incenter, Incircle, Perpendicular, Tangent.

Problem 559.
Triangle, Cevian, Incenters, Angle, 90 Degrees.

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

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

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

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

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

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

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

Sangaku Geometry Theorem: Cyclic Quadrilateral.
Incenters, Incircles, Inradii, Rectangle.

Sangaku Problem (An Old Japanese Theorem).
Incenters, Incircles, Inradii, Carnot's theorem.

Equal Incircles Theorem: Triangles, cevians, circles.

Adams' Circle Theorem
Concentric to the Incircle. 

Spieker Center
Incenter of the medial triangle.

Heron's Formula.
Key facts and a purely geometric step-by-step proof.
Incenter, Incircle, Excenter.
 

Semiperimeter and incircle of a triangle.

Semiperimeter, incircle and excircles of a triangle.

Gergonne Point Theorem.
Incenter, Incircle, Concurrency. Interactive proof with animation.

Key concept: Ceva's Theorem.
 

The Bevan Point The circumcenter of the excentral triangle.
Illustration with animation.

Interactive Gergonne Line and Nobbs Points. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Incenter and the points of contact of the incircle.

Triangle: Incircle, Perpendicular, Angle Bisector. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Prove proposition.

Right Triangles, Formulas and Facts
Pythagorean theorem, Similarity, Geometric mean, hypotenuse, cathetus (legs), semiperimeter, altitude, median, similar shapes areas, centers, orthocenter, incenter, incircle, centroid, circumcenter, radius, inradius.

Feuerbach Point Facts
Incenter, Incircle, Nine point circle, Circumcircle. 

Feuerbach Theorem, Acute Triangle illustration
 

Feuerbach Theorem, Obtuse Triangle illustration
 

Feuerbach Theorem, Right Triangle illustration
 

Feuerbach Theorem
One of the most striking theorems in triangle geometry.

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

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

Proposed Problem 260.
Equilateral Triangle, Incircle, Tangency Points, Vertices, Distances, Squares.

Proposed Problem 259.
Equilateral Triangle, Incircle, Tangency Points, Side, Distances, Squares.

Proposed Problem 258 Incircle.
Equilateral Triangle, Circumcircle, Point, Vertices, Side, Distances, Squares.

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

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

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

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

Proposed Problem 216 Incenter, Excenter. Quadrilateral, Angle Bisectors, and Concurrency.

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

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

Proposed Problem 209. Triangle, Incircles, Inradius.

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

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

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

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

Proposed Problem 200. Right Triangle, Incircle, Excircles, Points of Tangency, Inradius.

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

Proposed Problem 196. Triangle, Incircle, Inradius and Exradii Formula.

Journey to the Center of a Triangle (1976). Incenter, Circumcenter, Centroid, Orthocenter.

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

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

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

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

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

Proposed Problem 160. Triangle, Incircle, Incenter, Circumcircle, Circumcenter, Inradius.  

Proposed Problem 159. Distances from the Circumcenter to the Incenter and the Excenters.  

Proposed Problem 158. Relation between the Circumradius, Incircle, Inradius and Exradii of a triangle.  

Proposed Problem 155. Euler's Theorem: Distance from the Incenter to the Circumcenter.  

Proposed Problem 154. Triangle, Incircle, Inradius, Circumradius, Chord.

Proposed Problem 145. Four Triangles, Incircle, Tangent and Parallel to Side, Incenters, Circumcenters.  

Proposed Problem 144. Four Triangles, Incircle, Tangent and Parallel to Side, Inradii.

Proposed Problem 143. Four Triangles, Incircle, Tangent and Parallel to Side, Circumradii.

Proposed Problem 142. Four Triangles, Incircle, Tangent and Parallel to Side, Areas.

Proposed Problem 141. Triangle, Incircle, Tangent , Parallel, Perimeters.

Proposed Problem 137. Orthic Triangle, Altitudes, Perpendicular, Incircle, Collinear Points, Parallelogram.  

Proposed Problem 134. Orthic Triangle, Altitudes, Angle Bisectors, Orthocenter, Incenter.  

Proposed Problem 128. Incenter of a Triangle, Angle Bisectors, Sum of Ratios.

Proposed Problem 127. Centroid and Incenter of a Triangle, Parallel, Proportions.

Proposed Problem 126. Incenter of Triangle, Angle Bisector, Proportions.

Proposed Problem 120. Area of triangle, incenter, excircles, tangent.

Proposed Problem 119. Area of triangle, incenter, excircle, tangent.

Proposed Problem 118. Area of triangle, incenter, excenter, tangent.

Proposed Problem 117. Area of triangle, incenter, excircles, tangent.

Proposed Problem 114. Area of triangle, incenter, incircle, excircle.

Proposed Problem 113. Area of triangle, incenter, incircle, excircle.

Proposed Problem 110. Area of Contact Triangle. Incenter, Incircle.

Proposed Problem 96. Similar Triangles, Incenters, Parallelogram.

Proposed Problem 95. Similar Triangles, Inradii, Parallel.

Intouch and Extouch Triangles. Puzzle cut: 20 Piece Classic
Based on Proposed Problem 86.

Proposed Problem 86. Intouch and Extouch Triangles, Areas.

Contact Triangles. Puzzle cut: 22 Piece Polygons
Based on Proposed Problem 85.

Proposed Problem 85. Contact Triangles Areas, Incircle, Excircle.

Proposed Problem 84. Contact Triangles Areas, Incircle, Excircle, Inradius, Exradius.

Proposed Problem 82. Area of the Contact Triangle, inradius, circumradius.

Proposed Problem 81. Area of a triangle, side, incircle, incenter, inradius, circumradius.

Proposed Problem 80. Area of a triangle, side, incircle, incenter,, inradius.

Proposed Problem 68: Triangle, Incircle, Inradius, Tangent, Similarity.

Proposed Problem 64: Triangle, Incircle, Transversal.

Proposed Problem 52: Triangle, angles, incircle.

Puzzle: Sangaku Problem. Incircles, incenters,

Proposed Problem 40.
Triangle, Incenter, Excenter, Angles 80, 40, Distances.

Proposed Problem 39.
Triangle, Incenter, Incircle, Bisector, Cyclic Quadrilateral and angles.

Proposed Problem 38.
Right Triangle, altitude, incircles, incenters, and angles.

Proposed Problem 37.
Right Triangle, altitude, incircles, incenters, and orthocenter.

Proposed Problem 36.
Right Triangle, altitude, incircles and inradii.

Proposed Problem 35.
Incenters and Inradii in Cyclic Quadrilateral.

Proposed Problem 34.
Right triangle, cevian, incircles, tangents and inradius.

Problem 32.
Triangle, Cevian, Incircles, Tangents.

Problem 31. Right Triangle, Incircle, Collinears.
Problem 30. Right Triangle, Incircle, Inradius.

Proposed Problem 29.
Right Triangle, altitude, incircle and inradius. Ten conclusions.

Problem 29: Geometry Help.
Facts you should know.

Proposed Problem 28
Right Triangle, altitude, incircles and inradius.

Proposed Problem 27
Right Triangle, incircles and inradius.

Proposed Problem 26
Right Triangle, altitude, incircles and inradius.

Proposed Problem 25
Right Triangle, altitude, incircles and inradius.

Proposed Problem 24
Right Triangle, altitude, incircles and inradii.

Proposed Problem 23
Right Triangle, altitude, incircles and inradii.

Proposed Problem 22
Right Triangle, altitudes, incircles and inradii.

Proposed problem 20
Right Triangle, incircles and inradii.

Geometry Online Glossary .
Geometry Glossary based on the new New York State mathematics standards initiative. Incenter, Incircle.