The following two results display an interesting sort of 'duality' (symmetry) between the concepts of angle, incentre and angle bisector on the one hand, and those of side, circumcentre and perpendicular bisector on the other hand (De Villiers, 1994; 2009).
| 1) If I is the incentre of triangle ABC, then the circumcentre of triangle BIC, say O, lies on the angle bisector of angle A. | 2) If O is the circumcentre of triangle DEF, then the incentre of triangle EOF, say I, lies on the perpendicular bisector of side EF. |
Triangle Incentre-Circumcentre Collinearity
Challenge
Can you explain why (prove) the two results are true? Can you explain why (prove) them in more than one way?
If stuck, proofs of the two results are respectively given on p. 156-157 and p.176 in (De Villiers, 1994; 2009).
Some Related Links
Triangle Circumcircle Incentre Result
Japanese theorem for cyclic quadrilaterals
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Quadrilateral Similar Triangles Collinearity
Cyclic Hexagon Alternate Angles Sum Theorem
Circumscribed Hexagon Alternate Sides Theorem
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons: Generalizations of isosceles trapezia and kites
Angle Divider Theorem for a Cyclic Quadrilateral
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Tangential Quadrilateral Theorem of Gusić & Mladinić
Concurrency, collinearity and other properties of a particular hexagon
An interesting collinearity
More Properties of a Bisect-diagonal Quadrilateral
Euler-Nagel line Duality (analogy)
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
Concurrent Angle Bisectors of a Quadrilateral
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
An Inclusive, Hierarchical Classification of Quadrilaterals
Reference
De Villiers, M.D. (1994; 2009). Some Adventures in Euclidean Geometry (free to download as PDF). USA: Lulu Publishers.
(It is also available in printed bookform from Some Adventures in Euclidean Geometry (Print)).
Some External Links
Collinearity (Wikipedia)
Rectangle-rhombus duality (Wikipedia)
Incenter (Wikipedia)
Circumcircle (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers as well as books on problem solving can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, Created with JavaSketchpad, 7 April 2010: Updated to WebSketchpad, 12 May 2025.