Incenters in Cyclic Quadrilateral:
What is this about?
A Mathematical Droodle
8 April 2016, Created with GeoGebra
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Copyright © 1996-2018 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
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