IMC 2009 — Keystage 3 (IWYMIC) — individual Question 7

Quadrilateral $ABCD$ is not cyclic. Points $E,F,G,H$ are circumcentres of $ABD,ADC,BCD,ABC$, respectively. Lines $EG,FH$ meet at $I$, with $AI=4$, $BI=3$ cm. Find $CI$ in cm.