MathDB

6

Part of 2022 Oral Moscow Geometry Olympiad

Problems(2)

touchpoint of incircle, orthocenter and midpoint of altitude collinear wanted

Source: 2022 Oral Moscow Geometry Olympiad grades 8-9 p6

4/17/2022
In an acute non-isosceles triangle ABCABC, the inscribed circle touches side BCBC at point T,QT, Q is the midpoint of altitude AKAK, PP is the orthocenter of the triangle formed by the bisectors of angles BB and CC and line AKAK. Prove that the points P,QP, Q and TT lie on the same line.
(D. Prokopenko)
geometrycollinearcollinearity
equal faces of perpendicular edges in tetrahedron, concurrency related

Source: 2022 Oral Moscow Geometry Olympiad grades 10-11 p6

4/18/2022
In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges.
(Yu. Blinkov)
geometry3D geometrytetrahedronconcurrency