In an acute-angled triangle ABC the feet of the three altitudes are joined to form a smaller triangle, called the orthic triangle. Which point of triangle ABC is the incentre of that orthic triangle?

Let the altitude feet be D on BC, E on CA and F on AB. Because ∠BFC = ∠BEC = 90°, the points B, C, E and F are concyclic on the circle with diameter BC, and three more such circles appear in the same way. Chasing the equal angles those circles produce shows that each altitude of ABC bisects an angle of triangle DEF. The three altitudes meet at the orthocentre H, so H is where the three internal bisectors of the orthic triangle meet — that is, H is its incentre. The centroid and circumcentre of ABC have no such role, and the incentre of ABC lies on the bisectors of ABC, not of DEF.