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concyclic wanted, equal ratio of segments, altitudes related

Source: 2019 Austrian Federal Competition For Advanced Students, Part 2 p5

March 5, 2020

ratioequal ratioConcyclicgeometry

Problem Statement

Let

ABC

be an acute-angled triangle. Let

D

and

E

be the feet of the altitudes on the sides

BC

AC

. Points

F

and

G

are located on the lines

AD

and

BE

in such a way that

\frac{AF}{FD}=\frac{BG}{GE}

. The line passing through

C

and

F

intersects

BE

at point

H

, and the line passing through

C

and

G

intersects

AD

at point

I

. Prove that points

F, G, H

and

I

lie on a circle.
(Walther Janous)