Triangle Centers - Table of Content

Triangle Centers.

Incenter of a triangle. Index.

Excenter. Index.
Excircle, Exradius.

Circumcenter. Index.

Centroid /Barycenter. Index.

Orthocenter Index.

Nine-Point Center, Nine-Point Circle, Euler Line (English version).
Interactive illustration.

Circulo de los Nueve Puntos: Spanish version.
Interactive illustration.

Puzzle: Euler Line and Nine Point Center

Kosnita's Theorem.
Circumcenters, Concurrent lines.

Proposed Problem 370.
Triangle with squares, Circumcircles, Tangent circles.

Equal Incircles Theorem: Triangles, cevians, circles.

Proposed Problem 257.
Equilateral Triangle, Circumcircle, Point, Vertices, Side, Distances, Squares.

Proposed Problem 256.
Equilateral Triangle, Circumcircle, Point, Vertices, Distances.

Proposed Problem 255.
Triangle, Centroid, Vertices, a Point, Distances, Squares.

Proposed Problem 254.
Triangle, Centroid, Vertices, a Point, Distances, Squares.

Proposed Problem 253.
Triangle, Centroid, Vertices, Distances, Squares.

Proposed Problem 249.
Triangle sides, medians, 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 229. Triangle, Centroid, Transversal, Perpendicular lines.

Proposed Problem 226. Triangle, Centroid, Perpendicular lines.  

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

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

Proposed Problem 196. Triangle, Inradius and Exradii Formula.

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

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

Proposed Problem 209. Triangle, Incircles, Inradius.

Proposed Problem 208. Triangle, Excircles, Angles, 360 degrees.  

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

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

Proposed Problem 205. Right Triangle Area, Exradii relatives to legs or catheti.  

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

Proposed Problem 203. Right Triangle, Excircles, Exradii, Hypotenuse.  

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

Proposed Problem 201. Right Triangle, Excircles, Points of Tangency, Exradius, Semiperimeter.  

Proposed Problem 200. RightTriangle, Incircle, Excircles, Points of Tangency, Inradius.

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

Proposed Problem 196. Triangle, Inradius and Exradii Formula.

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

Proposed Problem 194. Area of a Triangle, Semiperimeter, Exradius.

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

Proposed Problem 191. Triangle, Altitudes, Orthocenter, Squares, Areas.  

Nagel Point Theorem. Proof.
 

Puzzle of the Nagel Point: 22 pieces of polygons.

Nagel Point Theorem - Flowchart Proof .
 

Semiperimeter and excircles of a triangle.

Semiperimeter and incircle of a triangle.

Semiperimeter, incircle and excircles of a triangle.

Gergonne Point Theorem. Concurrency. Interactive proof with animation.

Key concept: Ceva's Theorem.
 

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, Inradius and Exradii of a triangle.  

Proposed Problem 157. Distance from the Circumcenter to the Excenter.  

Proposed Problem 156. Triangle, Circumradius, Exradius, Chord, Secant line.  

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

Proposed Problem 154. Triangle, 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 140. Triangle, Excircle, Tangent, Semiperimeter.

Proposed Problem 139. Triangle Area, Orthic Triangle, Semiperimeter, Circumradius.  

Proposed Problem 138. Nagel's Theorem, Orthic Triangle, Altitudes, Circumradius, Perpendicular.  

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

Proposed Problem 136. Orthic Triangle, Altitudes, Perpendicular, Concyclic Points.  

Proposed Problem 135. Orthic Triangle, Altitudes, Perpendicular, Parallel  

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

Interactive Gergonne Line and Nobbs Points. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.

Triangle, Medians, Six Circumcenters Concyclic. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Prove proposition.

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

Proposed Problem 133. Triangle, Angle Bisectors, Collinear Points.

Proposed Problem 132.
Triangle, 60 degree, Orthocenter, Congruence, Midpoint.
 

Triangle with the bisectors of the exterior angles. Collinearity. Key concept: Menelaus Theorem.
 

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 116. Area of triangle, excircles, tangent.

Proposed Problem 115. Area of triangle, excircles, tangent.

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

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

Complete Quadrilateral: Ortholine-Steiner Line.
Using TracenPoche Dynamic Software
Step-by-Step construction, Manipulation, and animation

Proposed Problem 96. Similar Triangles, Incenters, Parallelogram.

Proposed Problem 95. Similar Triangles, Inradii, Parallel.

Proposed Problem 94. Similar Triangles, Circumcircles, Circumradii.

Proposed Problem 93. Similar Triangles, Circumcircles, Parallelogram.

Proposed Problem 92. Similar Triangles, Circumcircles, Circumradii, 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 83. Area of the Excircle Contact Triangle, exradius, circumradius.

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

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

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

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

Interactive Orthopole of a Triangle: Using TracenPoche Dynamic Geometry Software, Online Step-by-Step construction, manipulation, and animation.

Morley's Triangle & Center: with interactive animation and manipulation.

Schiffler Point: Four Euler Lines with interactive animation and manipulation.

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

Proposed Problem 67: Triangle, Circumcircle, Angles, Cyclic Quadrilateral.
 

Proposed Problem 66: Triangle, Excircle, Tangents, Geometric Mean.
 

Proposed Problem 64: Triangle, Incircle, Transversal.

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.

Sangaku Geometry Theorem: Cyclic Quadrilateral. Inradii, Rectangle.

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

Triangle Centers. Word find puzzle.

Proposed Problem 52: Triangle, angles, incircle.

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

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

Feuerbach Point Facts
 

Feuerbach Theorem, Acute Triangle illustration
 

Feuerbach Theorem, Obtuse Triangle illustration
 

Feuerbach Theorem, Right Triangle illustration
 

Proposed Problem 51: Fagnano's Problem
Inscribed Triangle with the Minimum Perimeter.

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

Proposed Problem 40. Geometry Help.
 

Adams' Circle Theorem
 

Spieker Center

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

Proposed Problem 39 Geometry Help.
Facts you should know for the proposed problem 39.

Second Ajima-Malfatti Point
Interactive illustration.

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

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

First Ajima-Malfatti Point
Animated illustration.

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

Lemoine Theorem

Triangle and Squares 17

Triangle and Squares 16

Sawayama -Thebault's theorem

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.

Heron's Formula with medians 

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.

Steiner Point
 

Mittenpunkt Point
 

Proposed problem 21
Acute triangle, orthocenter, diameter, tangents.

Proposed problem 20
Right Triangle, incircles and inradii.

Proposed problem 19
Right Triangle and Excenter

Symmedian (Lemoine) Point
Triangle Center

Heron's Formula.
Key facts and a purely geometric step-by-step proof.
 

Napoleon's Theorem. A purely geometric proof. It uses the Fermat point to prove Napoleon without transformations.
 

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

The Bevan Point: Puzzle