Diverticulosis: This is the most likely diagnosis given the presentation of profuse massive PR (per rectal) bleeding and the finding of a hyperemic and radish-like area in the transverse colon on colonoscopy. Diverticulosis is a common condition in adults over 40, and it can cause significant bleeding due to the rupture of a diverticulum.

Crohn's Disease: While less likely than diverticulosis for the specific symptom of massive PR bleeding, Crohn's disease can cause inflammation and ulcers in the colon, potentially leading to bleeding. The description of a hyperemic area could be consistent with active inflammation.
Ulcerative Colitis (UC): UC is another form of inflammatory bowel disease that could present with bleeding, although it typically starts in the rectum and extends proximally in a continuous fashion. The finding of a localized "radish-like" area might be less typical for UC.

Aortoenteric Fistula: Although rare and less likely, an aortoenteric fistula is a potentially life-threatening condition that must be considered in the differential diagnosis of gastrointestinal bleeding. It involves an abnormal connection between the aorta and the intestine, which can cause massive bleeding. Given the severity of the bleeding described, it's crucial not to miss this diagnosis.

Vascular Malformations or Tumors: These could include conditions like angiodysplasia or gastrointestinal stromal tumors (GISTs), which are less common causes of gastrointestinal bleeding. They might present with similar colonoscopic findings and could be considered if more common causes are ruled out.
Ischemic Colitis: This condition, caused by reduced blood flow to the colon, can lead to inflammation and potentially bleeding. However, it would be less likely to present with the specific finding of a "radish-like" area in the transverse colon.