Gardaí investigating the murder of national school teacher Ashling Murphy have carried out searches of similarities in Dublin and Co Offaly.
Both similarities are understood to be connected to a man believed to be the chief speculate in the 23-year-old’s murder.
Ms Murphy was killed by strangulation as she was jogging along a stretch of canal near Tullamore on Wednesday afternoon.
The man attended a Dublin hospital on Thursday in the company of a family member where he received treatment for what doctors initially believed were self-inflicted wounds.
He remains in hospital under Garda supervision while detectives await medical clearance to interview him.
It is not clear when he will be physically and mentally assessed for interview but gardaí were hopeful this will happen imminently.
The Irish Times understands this speculate has lived in south Dublin and in the midlands, where he has family members.
His family members have told gardaí they do not believe he is capable of committing such a crime.
One character in Co Offaly, near Tullamore, has been searched. Gardaí in Dublin have also searched another house in south Dublin.
Gardaí are confident they are nearing a breakthrough in the investigation and that DNA evidence will prove crucial in their inquiries.
Gardaí believe Ms Murphy, a talented musician from the townland of Blueball just outside Tullamore, tried to fight off her attacker, resulting in the man who killed her leaving forensic evidence at the scene.
Forensic evidence was used to exonerate a man arrested on Wednesday in Tullamore and he was released without charge late on Thursday night.
Gardaí have examined the crime scene, a Falcon Storm mountain bike they speculate was used by the killer, Ms Murphy’s remains and the clothes she was wearing when she was killed.
They believe when the forensic results yielded from those examinations are cross-checked with any suspects it will prove instrumental in catching the killer.
The Falcon Storm bike they believe was used by the killer on their way to the scene was recovered after and is now a crucial piece of evidence in the case.
The killer was also seen by a number of people at the scene and statements have been taken from those eyewitnesses.
Gardaí are investigating a complaint from another woman who has told them she was followed on the canal tow path where Ms Murphy was killed about two hours before the murder.
Gardaí are also examining CCTV footage recorded around Tullamore on Wednesday looking for images of anyone riding the Falcon mountain bike.
supplies expressed satisfaction with how that part of the investigation was progressing. Gardaí have appealed to anyone who travelled between Tullamore town centre and the wider approach roads to Grand Canal Way around Cappincur on Wednesday to come forward.
They are especially keen to speak to anyone with footage of road users, including CCTV, dashcam, GoPro or mobile phone images.
Thousands of people attended vigils for Ms Murphy on Friday and over the weekend across the country. There was also a vigil held at the London Irish Centre in Camden on Saturday afternoon.
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