Question: What was given its name due to its location directly beneath the Old Man of the Mountain, and is located in the state park that has the Flume Gorge and visitor center?

Context paragraphs:
Hocking Hills State Park: Hocking Hills State Park is a non-contiguous state park in the Hocking Hills region of Hocking County, Ohio, United States; it is contiguous with the Hocking State Forest.  Within the park are several spectacular rock formations, including Old Man's Cave, a narrow, deep gorge featuring waterfalls; Rock House, a cliffed area with a rock shelter; Cantwell Cliffs, a broad gorge at the head of a hollow with a unique stone stairway; and Ash Cave, a large rock shelter with a small waterfall, and Cedar Falls, a larger waterfall south of Old Man's Cave.

Rio Grande Nature Center State Park: Rio Grande Nature Center State Park is a state park of New Mexico, USA, located on the Rio Grande in Albuquerque.  A visitor center designed by architect Antoine Predock in 1982 contains exhibits on the area's riparian environment and a glass-walled library that overlooks a wildlife pond.  Two trails lead into the cottonwood forest (or "Bosque") from the visitor center.

Lake Fort Smith State Park: Lake Fort Smith State Park is a 260 acre Arkansas state park in Crawford County, Arkansas in the United States.  Originally a Fort Smith city park in the 1930s and later the Works Progress Administration-built Mountainburg Recreational Facility, the lake nestled in the Boston Mountains was adopted into the state park system by the Arkansas Department of Parks and Tourism in 1967.  Lake Fort Smith State Park was closed in 2002 to make way for a larger dam and spillway.  The addition flooded the site of the old park, and the new 260 acre Lake Fort Smith State Park reopened May 21, 2008 four miles north of its original location with 30 camp sites, 10 cabins, a group lodging facility, picnic sites, a pavilion, marina with rental boats, a double lane boat ramp, a swimming pool, playground, and an 8,000 square foot visitor center with exhibit gallery, gift shop, a meeting/class room, a patio with an outdoor wood burning fireplace, and an expansive view of the lake and mountains.

Flume Gorge: The Flume Gorge is a natural gorge extending 800 ft horizontally at the base of Mount Liberty in Franconia Notch State Park, New Hampshire, United States.  Cut by the Flume Brook, the gorge features walls of Conway granite that rise to a height of 70 to and are 12 to apart.  Discovered in 1808 by 93-year-old "Aunt" Jess Guernsey, the Flume is now a paid attraction that allows visitors to walk through the gorge.

Bushy Run Battlefield: Bushy Run Battlefield Park is a historical park that is operated by the Pennsylvania Historical and Museum Commission and the Bushy Run Battlefield Heritage Society on 218 acre in Penn Township, Westmoreland County, Pennsylvania in the United States.  It was the site of the Battle of Bushy Run fought on August 5–6, 1763 during the Pontiac's Rebellion.  The battle was a major victory for the British and enabled them to secure their control of the Ohio River Valley and what was to become the Northwest Territory.  Bushy Run Battlefield Park was established as a Pennsylvania State Park in the 1920s and became a National Historic Landmark in 1960.  The Visitor Center is open Wednesday through Saturday from 9:00 am until 5:00 pm, and Sunday from noon to 5:00 pm, during the months of May–October.  The Visitor Center hosts a museum exhibit entitled, "The March to Bushy Run", a theater, a gift shop and battlefield tours.  Bushy Run Battlefield Park is the only historic site or museum that deals exclusively with Pontiac's Rebellion.  Battle reenactments are held annually on the first full weekend of August.  The park is on Pennsylvania Route 993 near Harrison City and Jeannette.

Profile Lake: Profile Lake is a 13 acre water body located in Franconia Notch in the White Mountains of New Hampshire, at the foot of Cannon Mountain.  The lake was given its name due to its location directly beneath the Old Man of the Mountain, a famous rock formation which collapsed in 2003.  The lake is near the height of land in Franconia Notch; the lake's outlet is the Pemigewasset River, which flows south to the Merrimack River and ultimately the Gulf of Maine (Atlantic Ocean) at Newburyport, Massachusetts.

Quarry Visitor Center: Quarry Visitor Center, in Dinosaur National Monument in Utah was built as part of the National Park Service's Mission 66 program of modern architectural design in the US national parks.  This visitor center exemplifies the philosophy of locating visitor facilities immediately at the resource being interpreted.  The visitor center was closed from 2006 to 2011 due to structural damage from unstable soils.  The rotunda structure was demolished and replaced with a new structure of different design, while the quarry section was being stabilized and repaired.

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.

Profile House: The Profile House was a grand hotel in the White Mountains of New Hampshire, in the United States.  Originally built in 1852 and opening for its first season in 1853, it was operated by several owners and partners until its final season under the ownership of Karl P. Abbott, when the hotel, at its seasonal peak, burned to the ground, leaving only the train depot standing in the fire's aftermath.  Area attractions included Franconia Notch, the Great Boulder flume (Flume Gorge), Artist's Bluff, Mount Cannon, Profile Lake, Echo Lake, and Eagle Cliff.  The Profile House boasted amenities such as running water, electricity and all of the comforts to which the affluent guests had become accustomed.  The hotel was named for the iconic rock structure discovered by surveyors in 1805, that came to be known as Old Man of the Mountain.

Chesterfield Gorge Natural Area: Chesterfield Gorge Natural Area, also known as Chesterfield Gorge State Wayside, is a 13 acre state park in Chesterfield, New Hampshire, on Route 9.  The park conserves the rocky gorge with waterfalls on Wilde Brook, with a trail on either side.  There are picnic tables and a seasonal visitor center.

Supporting paragraphs: Franconia Notch State Park Profile Lake
Answer: Profile Lake