Let $H$ be the orthocenter of the equilateral triangle $\triangle ABC$. We know the distance between the orthocenters of $\triangle AHC$ and $\triangle BHC$ is $12$. What is the distance between the circumcenters of $\triangle AHC$ and $\triangle BHC$?

The number of revolutions of two pulleys is inversely proportional to their diameters. if a 24-inch diameter pulley making 400 revolutions per minute is belted to an 8-inch diameter pulley, find the number of revolutions per minute of the smaller pulley.

1) what is the measure of the exterior angle, ∠uvw ? 2) in triangle man, what is the measure of the exterior angle at n (in degrees)?

Three baby penguins and their father were sitting on an iceberg 0.50.50, point, 5 meters above the surface of the water. the father dove down 4.74.74, point, 7 meters from the iceberg into the water to catch dinner for his kids. what is the father penguin's position relative to the surface of the water?