Online Geometry

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

Incenter of a triangle - Table of Content

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.

Incenter by Folding

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

Distance between triangle centers

Distances between Triangle Centers Index.

Gergonne Point Theorems and Problems Index

Gergonne Points Index
Triangle Center

Nagel Point Theorems and Problems Index

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 to the centroid

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

Incenter, Intersecting circles

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

Tangent Circles, Incenter, Chord

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

Triangle, Incircle, Diameter

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

Outer Nagel Point

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

Gergonne Point Theorem

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

Triangle, 60 Degrees

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

Triangle, Circumcircle, Incenter

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

Right triangle, Incircles

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

Triangle, 60 Degrees, Centers, Angles

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

Triangle Centers, Angles

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

Triangle, Incircle, Parallel

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

Right triangle, Altitude, Incenter, Area

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

Triangle, Incenter, Excenter, Distance

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

Triangle, Incenter, Excenter, Similarity

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

Triangle, Incenter, Excenter, Concyclic Points

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

Triangle Area, 30 Degrees, Incircle

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

Triangle area, 60 degrees

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

Right triangle area

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

Triangle, Incircle, Altitude, Incenter, Angle

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

Triangle, Incircle, Inradius, Midpoint, Concurrency

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

Triangle, Incircle, Inradius, Altitude, Midpoint

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

Triangle, Incenter, Midpoint, Area

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

Triangle, Incenter, Midpoint, Perpendicular

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

Triangle, Incenter, Midpoints

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

Triangle, Incenter

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

Quadrilateral, Diagonals, Incenters

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

Geometry problem 567

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

Complete quadrilateral, Circles

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

Math Education: Triangle, Incenter

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

Triangle, Cevian, Incenters, 90 Degrees

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

Triangle, Cevian, Incircles

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

Triangle, Incenter, 90 Degrees

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

Circle, Angle, Incenter, Isosceles triangle

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

Triangle, CIrcle, Incircle, Inradius

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

Triangle, Incircle, Excircle, Inradius, Exradius

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

Triangle, Incircle, Inradius, Tangent

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

Triangle, Incenter, Parallel to a side

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

Tiangle, Incenter, Angle

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

Triangle, Circumcenter, Incenter, Excenter, Collinear

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

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.

Gergonne Line

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

Interactive Geometry

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.

Elearning 279: Tangent circles

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

Sagitta Problem

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

Elearning 260: Problem about equilateral triangle

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

Elearning 259: Equilateral triangle, Incircle

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

Elearning 258: Equilateral triangle, Incircle

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

Elearning: Napoleon Theorem III

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

Napoleon Theorem II

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

Napoleon Theorem I

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

Right triangle, inradius, altitude, arithmetic mean

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

Elearn: Quadrilateral, Angle Bisectors, Concurrency

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

Elearn 215: Cyclic quadrilateral

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

Elearn 213 Triangle Inradius, Common Tangents

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

Elearn 209

Proposed Problem 209. Triangle, Incircles, Inradius.

Problem 207: Hypotenuse

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

Elearn 206

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

Elearn 204

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

Problem 202 Right triangle, excircles

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

Elearning 200: Right Triangle

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

Elearning 197: Triangle Area

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

Elearning 196: Inradius, Exradii

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

Triangle Centers Video

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

Geometry: Triangle Area, inradius, exradii

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

Area of a triangle, semiperimeter, inradius

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

Elearning 187: Right triangle, incircles, angles

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

Elearning 186: Right triangle, circles

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

Elearning 186: Right triangle, circles

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

Elearning: Triangle, Circle, Incircle, Radius

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

Triangle Elearning 159

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

Geometry Problem 158

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

Elearning 155

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

Elearning 154

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

Geometry Problem 145. Elearning Triangle

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

Geometry problem 144 Triangle

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

Geometry problem 143

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

Elearning 142: Triangle area

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

Elearning 141

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

Geometry Problem 137

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

Geometry Problem 134

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

Elearn 128: Incenter, Ratios

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

Elearn 127: Centroid, Incenter

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

Elearn 126: Incenter Proportions

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

Problem 120: Triangle area. Elearning.

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

Elearn 119: Triangle area.

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

Problem 118

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

Elearning 117 Areas

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

Elearning 114: Area

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

Elearning 113: Areas

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

Contact Triangle Area. Elearning.

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

Elearning 96

Proposed Problem 96. Similar Triangles, Incenters, Parallelogram.

Problem 95, Elearning

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.

Contact triangle area

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

Triangle area

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

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 terms

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

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