MathDB

2

Part of 2022 Oral Moscow Geometry Olympiad

Problems(2)

triangle construction given intersectionf of <B,<C bisectors and axis of BC

Source: 2022 Oral Moscow Geometry Olympiad grades 8-9 p2

4/17/2022
Angle bisectors from vertices BB and CC and the perpendicular bisector of side BCBC are drawn in a non-isosceles triangle ABCABC. Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler.
(Yu. Blinkov)
geometryconstruction
concyclic wanted, tangents of circumcircle related

Source: 2022 Oral Moscow Geometry Olympiad grades 10-11 p2

4/18/2022
In an acute triangle ABCABC,OO is the center of the circumscribed circle ω\omega, PP is the point of intersection of the tangents to ω\omega through the points BB and CC, the median AM intersects the circle ω\omega at point DD. Prove that points A,D,PA, D, P and OO lie on the same circle.
(D. Prokopenko)
geometryConcyclic