Suppose triangle ABC has perimeter p and three semicircles with diameter AB, BC and CA are drawn on the outside of triangle ABC. Let H be a semicircle that contains the three semicircles. Prove that the radius is at least p/4.
The sum of all three small radii is p/2.
The large radius must be grater than the sum of any two small radii.
The Triangle Inequality Theorem states the sum of any two sides of a triangle is greater than the third side. This is equivalent to saying the sum of any two sides is greater than the semiperimeter.
Then the sum of any two small radii must be greater than half the semiperimeter; half of the semiperimeter is p/4.
Then the large radius must also be greater than p/4.