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 circumcentre of triangle EOF, say I, lies on the perpendicular bisector of EF.

Triangle Incentre-Circumcentre Collinearity

Can you explain why (prove) the two results are true? If stuck, proofs of the two result are respectively given on p. 156-157 and p.176 of Some Adventures in Euclidean Geometry, which is available for purchase as downloadable PDF or printed book at More Info