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Two first-half goals from Tom Rogic and Scott Sinclair were cancelled out in the closing stages by Inverness’ late equaliser.
Rogic opened the scoring for Celtic on 17 minutes after getting on the end of Kieran Tierney’s terrific cross.
Inverness equalised through Billy King after 28 minutes with a stunning strike after cutting inside from the right wing.
Question marks will be asked of Dorus de Vries once again, who got a hand to the ball but couldn’t prevent it from hitting the back of the net.
Erik Sviatchenko was lucky he wasn’t sent off after a foul on Ross Draper outside the area that wasn’t penalised, leaving the home side feeling angered by the decision.
While Inverness were still lamenting the decision when Scott Sinclair put Celtic back in front three minutes later after winning back possession just inside his own half.
Following a driving run forward, Sinclair curled the ball beyond the ‘keeper becoming the first Celtic player since Jimmy McGrory to score in every league game so far.
Substitute Alex Fisher levelled the game with two minutes remaining.
Here are the player ratings from today’s game.