Garden Geometry 1588: Unlocking the Tangent: Prove the Fifth of the Radius in a Sector with Semicircle

Proving Tangency and Proportions in a Sector-Semicircle Configuration

Dive into the geometry of a circular sector combined with a semicircle. This problem challenges you to demonstrate a fascinating proportional relationship involving a tangent chord and the sector's radius. Analyze the setup, uncover the hidden connections, and prove that the length of the tangent segment is precisely one-fifth of the sector's radius. Perfect for geometry enthusiasts eager to explore elegant mathematical proofs.

Problem 1588 Statement

In a circular sector AOB with a 90-degree central angle, a semicircle centered at Q has OB as its diameter. The chord AD is tangent to the semicircle at C. Prove that CD is one-fifth of the sector's radius.

Garden Geometry problem 1588: Diagram of a circular sector with a semicircle and a tangent chord, showcasing an elegant geometric problem

Circles and tangents,
Secrets in the garden bloom,
Geometry's grace.

Uncover and share solutions to this problem.


Flyer of problem 1588 Designed with iPad Apps

Flyer of Geometry problem 1588  Diagram of a circular sector with a semicircle and a tangent chord, showcasing an elegant geometric problem