MathDB

6

Part of 2022 South East Mathematical Olympiad

Problems(2)

four points lie on one circle

Source: 2022 China Southeast Grade 10 P6

8/3/2022
HH is the orthocenter of △ABC\triangle ABC,the circle with center HH passes through AA,and it intersects with AC,ABAC,AB at two other points D,ED,E.The orthocenter of △ADE\triangle ADE is H′H',line AH′AH' intersects with DEDE at point FF.Point PP is inside the quadrilateral BCDEBCDE,so that △PDE∼△PBC\triangle PDE\sim\triangle PBC.Let point KK be the intersection of line HH′HH' and line PFPF.Prove that A,H,P,KA,H,P,K lie on one circle. https://i.ibb.co/mcyhxRM/graph.jpg
geometry
Three circles share a common point

Source: 2022 China Southeast Grade 11 P6

8/3/2022
Let OO be the circumcenter of △ABC\triangle ABC. A circle with center PP pass through AA and OO and OPOP//BCBC. DD is a point such that ∠DBA=∠DCA=∠BAC\angle DBA = \angle DCA = \angle BAC. Prove that: Circle (P)(P), circle (BCD)(BCD) and the circle with diameter (AD)(AD) share a common point. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS9jLzlmZjdlN2ExZDJjYjAwYWJlZTQzYWRkYzg3NDlhMTUyZjRlNGJjLmpwZw==&rn=c291dGhlYXN0UDYuanBn
geometry