Triangle Inscribing in a Circle

Select three points at random on a unit Circle. Find the distribution of possible areas. The first point can beassigned coordinates (1, 0) without loss of generality. Call the central angles from the first point to the second andthird and . The range of can be restricted to [0, ] because of symmetry, but can range from . Then

(1)

(2)

(3)

			(4)

But

	(5)

Write (4) as

(6)

(7)

(8)

			(9)

(10)

			(11)

(12)

(13)

			(14)

• Argument Addition Relation
• Argument (Complex Number)
• Argument (Elliptic Integral)
• Argument (Function)
• Argument Multiplication Relation
• Argument Principle
• Argument Variation
• Aristotle's Wheel Paradox
• Arithmetic
• Arithmetic-Geometric Mean
• Arithmetic Geometry
• Arithmetic-Harmonic Mean
• Arithmetic-Logarithmic-Geometric Mean Inequality
• Arithmetic Mean
• Arithmetic Progression
• Arithmetic Sequence
• Arithmetic Series
• Armstrong Number
• Arnold Diffusion
• Arnold's Cat Map
• Arnold Tongue
• Aronhold Process
• Aronson's Sequence
• Arrangement
• Arrangement Number