Geometry
# Triangle Centers

At the 1920 FIFA World Cup, there was an earth quake during one of the games and a cone shaped hole was created with a diameter of 4 feet and sloping sides, both 4 feet long from ground-level to the bottom of the pit. The soccer ball, with a radius of one foot, fell into the hole. Find the distance from the center of the Soccer ball to the bottom of the pit in feet.

**Assumptions:**

- The sides of the hole are perfectly straight and smooth.

- The ball falls as far as it can with out changing shape.

- FYI, there wasn't actually an earthquake at the 1920 FIFA World Cup.

In triangle \(ABC\) with centroid \(G\), if \( AG=BC\), what is angle \(BGC \) in degrees?

Note: Diagram is not drawn to scale.

A triangle has sides of 6, 8, and 10 inches.

What is the distance between incenter and circumcenter of the triangle?

In the diagram above, line \(l\) passes through the centroid of \(\triangle ABC.\)

If the perpendicular distance between \(A\) and line \(l\) is 2, and the perpendicular distance between \(B\) and line \(l\) is 6, then what is the perpendicular distance between \(C\) and line \(l?\)

Consider an isosceles \(\triangle ABC\) with \(AB=AC=5, BC=6,\) where \(I,O,H\) denote its incenter, circumcenter, orthocenter, respectively.

Find the area of \(\triangle IOH\).