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Mr Thorpe responded to questions from the media for an hour after delivering a prepared statement in a bid to clear his name.
The former Liberal leader, who was accompanied by his wife Marion, made a point-by-point denial covering all aspects of the affair.
He began by talking about his first acquaintance with Mr Scott in 1961 and went on to talk about the alleged plot in which former airline pilot Andrew Newton is supposed to have been offered £5,000 to kill Mr Scott.
Mr Newton made the claim recently in two London evening newspapers.
The pilot had been jailed for two years in 1976 after shooting Mr Scott's dog on a lonely stretch of Exmoor, but at the time there was no connection made with Mr Thorpe.
But Mr Newton was recently quoted in the London Evening News, in the presence of his solicitor, as saying that it was a "contract to murder" and he was offered money by a leading Liberal to kill Mr Scott.
Mr Thorpe denied any involvement in the alleged plot and said that he had a brief "affectionate friendship" with the former male model in the 1960s and "no sexual activity of any kind took place".
He added: "I was at no time party to any form of cover-up or attempt to put pressure on Scott."
Mr Thorpe also said that he did not know Mr Newton and that he has had no direct or indirect communication with him.
The former Liberal leader denied discussing with ex-Liberal MP Peter Bessell or with anyone else any proposed murder plot or attempt to harm Mr Scott and he said he had no intention of resigning.
He said that the whole affair had put him and his family under a great deal of pressure.
Mr Thorpe added: "It would be insane to pretend that the re-emergence of this story has not placed an almost intolerable strain on my wife, my family and on me.
"Only their steadfast loyalty and the support of many friends known and unknown all over the country has strengthened my resolve and determination to meet this challenge."
Mr Thorpe said it was neither for him nor the press to investigate any alleged murder plot and it was a matter for the police.
The Norman Scott case eventually went to court in 1979.
Jeremy Thorpe, deputy treasurer of the Liberal Party David Holmes and two other men - John le Mesurier and George Deakin - were charged and cleared of conspiracy to murder charges against Norman Scott after a month.
Although he was cleared, Mr Thorpe lost his seat in the general election leading up to the case and his career was ruined by the whole affair.
He had held the parliamentary seat for 20 years.
Mr Thorpe was eventually put forward for peerage in 1999 but opinions were divided because it was felt he was disgraced and he would not be an effective member of the House of Lords because he was suffering with Parkinson's disease.
He published his memoirs later that year, saying that he never had any doubt about the acquittal of all the defendants on trial.
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