MathDB

4

Part of 2005 Oral Moscow Geometry Olympiad

Problems(2)

FD _|_ BE wanted, hexagon ABCDE, AB = BC, CD = DE, EF = FA, <A=<C=90^o

Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p4

10/16/2020
Given a hexagon ABCDEFABCDEF, in which AB=BC,CD=DE,EF=FAAB = BC, CD = DE, EF = FA, and angles AA and CC are right. Prove that lines FDFD and BEBE are perpendicular.
(B. Kukushkin)
geometryperpendicularhexagonequal segmentsright angle
aphere inscribed into pyramid, with parallelogram base

Source: 2005 Oral Moscow Geometry Olympiad grades 10-11 p4

10/21/2020
A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal.
(M. Volchkevich)
geometryparallelogrampyramidsphere