MathDB

GJ , FB , HK concurrent or parallel , ABCD cyclic and orthodiagonal

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E+4

November 29, 2022

geometryconcurrencyconcurrentCyclic

Problem Statement

ABCD

is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let

O

denote the centre of its circumcircle and

E

the intersection of the diagonals.

J

and

K

denote the perpendicular projections of

E

on the sides

AB

and

BC

. Let

F , G

and

H

be the midpoint line segments. Show that lines

GJ

FB

and

HK

either pass through the same point or are parallel to each other.