Incenters in Cyclic Quadrilateral:
What is this about?
A Mathematical Droodle
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander BogomolnyIncenters in Cyclic Quadrilateral
Every (convex) quadrilateral ABCD defines four triangles: BCD, ACD, ABD, and ABC. The applet is an attempt to convey an assertion often referred to as the Japanese theorem:
The incenters of the four triangles form a rectangle.
Proof
From ΔBCD, ∠BI_{BCD}C = 90° + ∠BDC/2.
Similarly, from ΔABC, ∠BI_{ABC}C = 90° + ∠BAC/2.
However, since we assumed that the quadrilateral ABCD is cyclic, ∠BDC = ∠BAC. Therefore, also
(1)  ∠BCI_{BCD} + ∠BI_{ABC}I_{BCD} = 180°. 
Similarly the quadrilateral AI_{ABD}I_{ABC}B is cyclic, so that
(2)  ∠BAI_{ABD} + ∠BI_{ABC}I_{ABD} = 180°. 
(1) and (2) imply that
∠BI_{ABC}I_{BCD} + ∠BI_{ABC}I_{ABD} = 360°  ∠BCD/2  ∠BAD/2 = 270°.
Which gives
(3)  ∠I_{ABD}I_{ABC}I_{BCD} = 90°. 
Other angles are treated similarly. Q.E.D.
In addition, let P, Q, R, and S be the midpoints of the arcs AB, CD, BC, and AD, respectively.
It's almost immediate to prove that
This is #3.5 from Fukagawa and Pedoe's collection. It was written on a 1880 tablet in the Yamagata prefecture. Maruyama Ryoukan posted a tablet in 1800 at the Sannosha shrine in Tsuruoka city of Yamagata prefecture with the observation that [Fukagawa and Rothman, p. 192]
r_{ABC} + r_{ACD} = r_{ABD} + r_{BCD}.
(In itself, this is a particular case of what is known as the Old Japanese Theorem.)
This implies that the lines through the incenters parallel to the diagonals form a rhombus.
You may observe that the arc bisectors of the arcs subtended by the sides of the quadrilateral serve as the diagonals of the rhombus.
References
 H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
Write to:
Charles Babbage Research Center
P.O. Box 272, St. Norbert Postal Station
Winnipeg, MB
Canada R3V 1L6  H. Fukagawa, A. Rothman, Sacred Geometry: Japanese Temple Geometry, Princeton University Press, 2008
 D. Wells, You Are A Mathematician, John Wiley & Sons, 1995
Sangaku

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Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny