MathDB

4

Part of 2008 Oral Moscow Geometry Olympiad

Problems(2)

concyclic projections wanted, inside a cyclic quadrilateral

Source: 2008 Oral Moscow Geometry Olympiad grades 8-9 p4

10/13/2020
A circle can be circumscribed around the quadrilateral ABCDABCD. Point PP is the foot of the perpendicular drawn from point AA on line BCBC, and respectively QQ from AA on DCDC, RR from DD on ABAB and TT from DD on BCBC . Prove that points P,Q,RP,Q,R and TT lie on the same circle.
(A. Myakishev)
geometryConcyclicprojectionscyclic quadrilateral
distance form circumcenter to orthocenter is AB+ AC when <A=120^o

Source: 2008 Oral Moscow Geometry Olympiad grades 10-11 p4

10/14/2020
Angle AA in triangle ABCABC is equal to 120o120^o. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to AB+ACAB + AC.
(V. Protasov)
geometryorthocenterCircumcenterdistance