Incenters in Cyclic Quadrilateral:
What is this about?
A Mathematical Droodle

8 April 2016, Created with GeoGebra

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Incenters 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.

Japanese theorem: rectangle in a cyclic quadrilateral

Proof

From ΔBCD, ∠BIBCDC = 90° + ∠BDC/2.

Similarly, from ΔABC, ∠BIABCC = 90° + ∠BAC/2.

However, since we assumed that the quadrilateral ABCD is cyclic, ∠BDC = ∠BAC. Therefore, also ∠BIBCDC = ∠BIABCC. The quadrilateral BIABCIBCDC is therefore cyclic, so that

(1) ∠BCIBCD + ∠BIABCIBCD = 180°.

Similarly the quadrilateral AIABDIABCB is cyclic, so that

(2) ∠BAIABD + ∠BIABCIABD = 180°.

(1) and (2) imply that

∠BIABCIBCD + ∠BIABCIABD = 360° - ∠BCD/2 - ∠BAD/2 = 270°.

Which gives

(3) ∠IABDIABCIBCD = 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.

arc bisectors in a cyclic quadrilateral

It's almost immediate to prove that PQ⊥RS. This is also true that PQ and RS are parallel to the sides of the rectangle of the incenters and pass through their midpoints.

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]

rABC + rACD = rABD + rBCD.

(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.

rhombus in a cyclic quadrilateral

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

  1. 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

  2. H. Fukagawa, A. Rothman, Sacred Geometry: Japanese Temple Geometry, Princeton University Press, 2008
  3. D. Wells, You Are A Mathematician, John Wiley & Sons, 1995

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