Sangaku for Incircles

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The Sangaku (old Japanese theorem) states that the centers of the incircles of the four triangles defined by the sides and the diagonals inside any concyclic quadrilateral are vertices of a rectangle.

Contributed by: Claude Fabre (March 2011)
Open content licensed under CC BY-NC-SA



It is quite unexpected that the centers of the incircles form a rectangle, no matter how the concyclic quadrilateral is distorted. For a proof, decompose the angles of the figure made by the centers. Find relations with the angles of the concyclic quadrilateral. Use properties of the angles that are due to the quadrilateral being concyclic. The proof is not trivial.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.