Lecture Notes: Solving a Geometric Problem with Circles

Problem Introduction

• Presenter: Presh Talwalker
• Objective: Determine the sum of the radii, (a + b + c), of inscribed circles within a larger circle with a radius of 12.
• Configuration:
  • Start with a circle of radius 12.
  • Inscribe two circles each passing through the center of the large circle with radius (b).
  • Inscribe a third circle tangent to the large circle and the two smaller circles with radius (a).
  • Finally, inscribe a fourth circle tangent to the diameter, the large circle, and one circle with radius (b) with radius (c).

Solving for Radius (b)

• Construction: Diameter of circle equals (2b).
• Since (2b) is a radius of the large circle:
  • (2b = 12)
  • (b = 6)

Solving for Radius (a)

• Triangle Construction:
  • One leg = (b)
  • Hypotenuse = (a + b)
  • Other leg = (12 - a)
• Using Pythagorean Theorem:
  • ((a + b)^2 = b^2 + (12 - a)^2)
  • Substitute (b = 6), simplify:
  • Solve to find (a = 4)

Solving for Radius (c)

• First Construction:
  • Large circle radius = 12
  • One leg = (c)
  • Hypotenuse = (12 - c)
  • Solve for (x^2 = (12 - c)^2 - c^2)
  • Simplify to (x^2 = 144 - 24c)
• Second Construction:
  • One leg = (x)
  • Other leg = (b - c)
  • Hypotenuse = (b + c)
  • Solve for (x^2 = (b + c)^2 - (b - c)^2)
  • Simplify to (x^2 = 4bc)
  • Substitute (b = 6), find (x^2 = 24c)
• Equate Equations:
  • Solve (24c = 144 - 24c)
  • Simplify to find (c = 3)

Conclusion

• Sum of Radii: (a + b + c = 4 + 6 + 3 = 13)

• Acknowledgement: Thanks to David for the suggestion.
• Closing Note: Encouragement to join the YouTube community for more problem solving.