Question: Lonesome Lake is located south of a 4080 ft peak that was home to what?

Context paragraphs:
Franconia Notch State Park: Franconia Notch State Park is located in the White Mountains in northern New Hampshire, United States, and straddles 8 mi of Interstate 93 as it passes through Franconia Notch, a mountain pass between the Kinsman Range and Franconia Range.  Attractions in the state park include the Flume Gorge and visitor center, the Old Man of the Mountain historical site, fishing in Echo Lake and Profile Lake, and miles of hiking, biking and ski trails.  The northern part of the park, including Cannon Mountain and Echo and Profile lakes, is in the town of Franconia, and the southern part, including Lonesome Lake and the Flume, is in Lincoln.

Imogene Peak: Imogene Peak, at 10125 ft above sea level is a peak in the Sawtooth Range of Idaho.  The peak is located in the Sawtooth Wilderness of Sawtooth National Recreation Area in Custer County.  The peak is located 1.83 mi north of Parks Peak, its line parent.  Farley Lake is located south of the peak, and Imogene Lake is west of the peak.

Lonesome Lake Peak: Lonesome Lake Peak, also known as Lonesome Peak, at 11302 ft above sea level is a peak in the White Cloud Mountains of Idaho.  The peak is located in Sawtooth National Recreation Area in Custer County.  The peak is located 2.07 mi south-southeast of D. O. Lee Peak, its line parent.  It is the 68th highest peak in Idaho and rises above the west side of Lonesome Lake, the uppermost of the Boulder Chain Lakes.

Cannon Mountain (New Hampshire): Cannon Mountain (formerly Profile Mountain) is a 4080 ft peak in the White Mountains of New Hampshire.  It is known for both its technical rock and ice climbing on its cliff face and skiing at Cannon Mountain Ski Area.  Until the formation collapsed on May 3, 2003, it was also home to the Old Man of the Mountain.

San Mateo Peak: San Mateo Peak is the unofficially named 3,591 ft peak, at the western end of the ridge running west then northwest from Elsinore Peak to Morrell Canyon, south and west of the Morrell Potrero in the Elsinore Mountains of the Santa Ana Mountain Range.  The peak name was given by Sierra Club Lower Peaks Committee Guide, originally named by Ken Croker who for 20 years maintained trails in the Santa Ana Mountains with volunteers from the Sierra Club.  It remains officially nameless despite being the highest summit in the Elsinore Mountains, higher than Elsinore Peak, at 3536 ft., the highest named peak in those mountains.

Kinsman Range: The Kinsman Range, also known as the Cannon–Kinsman Range, is located in the White Mountains of New Hampshire in the United States.  The highest point in the north-south range is 4358 ft Kinsman Mountain, followed by the 4293 ft North Peak of Kinsman, and 4080 ft Cannon Mountain, one of the richest in rock climbing routes in the Whites.  All are official "Four-thousand footers".  3478 ft Mount Wolf is on the crest of the range south of Kinsman Mountain.  Rounding out the range is 2470 ft Bald Peak on the west side of Kinsman Mountain.

Nokhu Crags: Nokhu Crags is a rock formation and mountain summit in the Never Summer Mountains range of the Rocky Mountains of North America.  The name is derived from the Arapaho language, "Neaha-no-xhu", meaning "Eagles Nest."  The 12490 ft peak is located in State Forest State Park, 4.0 km south (bearing 181°) of Cameron Pass in Jackson County, Colorado, United States.  The summit lies just northwest of the Continental Divide and Rocky Mountain National Park, near the headwaters of the Michigan River.  The peak is prominently visible from State Highway 14 and can be seen throughout the southern North Park basin where it is known also known as "the Crags" or "Sleeping Indian" for its resemblance to the form of a supine chief.  To the east lie the shallow basins of Snow Lake and the Michigan or American Lakes; to the north lies a snow filled couloir; to the west the mountain descends directly into the deep waters of Lake Agnes; and to the south lie Static Peak, Mount Richthofen, and the remainder of the Never Summer Mountain Range.

Matterhorn Peak: Matterhorn Peak is located in the Sierra Nevada, in the western U.S. state of California, at the northern boundary of Yosemite National Park.  At 12,285 ft elevation, it is the tallest peak in the craggy Alps-like Sawtooth Ridge and the northernmost 12000 ft peak in the Sierra Nevada.  The peak also supports the Sierra's northernmost glacier system.  It was named after the Matterhorn in the Alps.

Lonesome Lake (Idaho): Lonesome Lake is an alpine lake in Custer County, Idaho, United States, located in the White Cloud Mountains in the Sawtooth National Recreation Area.  The lake is accessed from Sawtooth National Forest trail 683.

Lonesome Lake (New Hampshire): Lonesome Lake is a 12.2 acre water body located in Franconia Notch in the White Mountains of New Hampshire, south of Cannon Mountain.  The lake is reachable by hiking trails from the Franconia Notch Parkway and features an Appalachian Mountain Club hut on its southwest shore.

Supporting paragraphs: Lonesome Lake (New Hampshire) Cannon Mountain (New Hampshire)
Answer: the Old Man of the Mountain