There are four Euclidean centres of a triangle--the circumcentre, the centroid, the incentre and the orthocentre. In this article, the authors prove the following: if the centre is the incentre (resp. orthocentre) then there exists a triangle with given distances of its vertices from its incentre (resp. orthocentre). They also consider uniqueness and constructibility of such triangles. (Contains 2 figures.)