Police say the cyclist then left the area only to return a short time later armed with two large axes that he pulled from his duffel bag.
He allegedly threatened the man with the axes before leaving and walking towards a shopping centre on Claremont St where he placed the weapons back in his bag.
But police say the man, and two of his relatives, chased the cyclist into the supermarket car park and tackled him to the ground.
The pair then became involved in a scuffle where the man had the tip of his finger bitten off.
The cyclist ran from the area but was arrested a short time later in Evaline St, Campsie by Public Order and Riot Squad officers.
He was taken to Royal Prince Alfred Hospital for assessment before being released into police custody about 10.50pm on Monday night.
He was charged with causing grievous bodily harm with intent, affray, armed with intent to commit indictable offence, common assault and drug possession.
He was remanded to appear at Burwood Local Court on Tuesday.
The injured man was treated at Bankstown Hospital where he will undergo surgery to his middle finger.
jack.paynter@news.com.au