Crowd trouble marred the end of the Scottish Cup final as Hibernian ended a 114-year wait to win the trophy.
Thousands of Hibs fans streamed on to the pitch at the final whistle at Hampden Park and fighting broke out when some Rangers fans then got past stewards.
Police horses were eventually brought on to restore order as police and stewards struggled to cope.
Hibs won the match 3-2 with a David Gray goal in injury time.
Rangers players appeared to be goaded by Hibs supporters as they tried to leave the pitch.
Hampden officials said the players would not return to the stand to collect their runners-up medals, receiving them in the changing room instead.
Police horses were last needed at the 1980 Scottish Cup final between Celtic and Rangers.
As Hibs fans returned to the stands part of the pitch was ripped up and the goal posts were broken by people climbing on them.
A statement from Rangers read: "Rangers Football Club is shocked at the disgraceful behaviour of Hibs fans at full time.
"Rangers players and staff were assaulted by these fans who invaded the pitch and in the interests of their safety could not return to the pitch for the medal presentation.
"There can be no place for the violent behaviour witnessed at the end of the Final and Rangers fully expect the Scottish Football Association and Police Scotland to launch an investigation to find out where security measures failed.
"The Club would like to commend Rangers fans for the restraint the showed under severe provocation."
Mr Regan said in a statement: "The Scottish FA is appalled by the scenes of disorder on the field immediately after full time and at the contemptible behaviour that ensued.
"What should have been an historic, memorable Scottish Cup final will now, sadly, be remembered for all the wrong reasons.
"Police Scotland and the stadium management initiated a crisis liaison group as soon as the disorder began and we thank them for containing the situation as quickly as possible.
"The Scottish FA, along with Police Scotland and the stadium management, have initiated a full investigation into the scenes that unfolded to understand how such a volume of supporters were able to enter the field."
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