Theorem
Given any triangle ABC, construct the angle bisector of angle BAC and extend to meet the circumcircle of ABC in F. Prove that F is the midpoint of arc BC, and if a circle with F as centre and FB as radius is constructed, then the point P, the intersection of this new circle and FA, is the incentre of triangle ABC.
Triangle Circumcircle Incentre Result
Challenge
Can you explain why (prove that) the result is true?
Application
This result is useful in proving these results: Cyclic Quadrilateral Incentres Rectangle as well as Cyclic Quadrilateral Angle Bisectors Rectangle Result.
Reference
A proof of the result is given on p. 190 of Some Adventures in Euclidean Geometry (free to download) or in printed bookform from Some Adventures in Euclidean Geometry (Print).
Some Related Links
Cyclic Quadrilateral Incentres Rectangle
Cyclic Quadrilateral Midpoints of Arcs Theorem
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Angle Sum 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
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Some Circle Concurrency Theorems
Converse of Tangent-Secant Theorem (Euclid Book III, Proposition 36)
Some Generalizations of Napoleon's Theorem
A Cyclic Quadrilateral Generalization of a Golden Rectangle property
A variation of Miquel's theorem and its generalization
Minimum Area of Miquel Circle Centres Triangle
Matric Exam Geometry Problem - 1949
A 1999 British Mathematics Olympiad Problem involving a Cyclic Hexagon
An extension of the IMO 2014 Problem 4
Six Point Cevian Circle
Nine Point Conic and Generalization of Euler Line
Geometry Loci Doodling with Cyclic Quadrilaterals
Crossed Quadrilateral Properties
Cyclic Kepler Quadrilateral Conjectures
Cross's (Vecten's) theorem & generalizations to quadrilaterals
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
Some further generalizations of an associated result of the Van Aubel configuration using pairs of similar triangles
Eight Point Conic for Cyclic Quadrilateral
Bradley's Theorem for a Circumscribed Quadrilateral
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
The quasi-circumcentre and quasi-incentre of a quadrilateral
Bicentric Quadrilateral Properties
Concurrent Angle Bisectors of a Quadrilateral
Triangle Incentre-Circumcentre Collinearity
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
An Inclusive, Hierarchical Classification of Quadrilaterals
Some External Links
Incenters in Cyclic Quadrilateral (Cut The Knot)
Cyclic quadrilateral (Wikipedia)
Cyclic Quadrilateral (Wolfram MathWorld)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers 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, Dec 2009; updated to WebSketchpad, 5 May 2025.