# Three Squares and Two Ellipses

Two congruent ellipses with the major axis 2a and the minor axis 2b touch each other, the minor axis of each passing through the point of contact; *l* is a common exterior tangent parallel to the minor axis.

There are three congruent squares of side t such that two of the squares have a side in common and another side lying on *l*, and each has a vertex on one of the ellipses, and a third square has a vertex on each ellipse and a side that lies along a side of the first two squares.

Show that

a = 2b and t = a/5.

The problem was carved on an 1838 tablet in the Aichi prefecture. The tablet has since disappeared, but the solution has been discussed in 1976 by H. Fukagawa with a reference to a 1837(!) publication, *Chosyu Sinpeki* by Tameyuki Yosida. The title translates as *Tablets in the Chosyu Prefecture*.

### References

H. Fukagawa, D. Pedoe,

*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989 (pp. 62, 154-155)Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

The calculations that are necessarily a part of any solution are greatly simplified by an affine trasnformation that squeezes the diagram towards *l* with coefficient b/a. The horizontal dimensions remain the same, whereas all vertical dimensions are mulitplied by b/a. In particular, the ellipses convert to the circles of radius b and the t×t squares become rectangles t'×t, with

In the diagram,

From

it follows that

(1) | 2bt²/a = b² - 2bt(1 + b/a) + t²(1 + b/a)² |

so that (very clever, I must say)

(1') | t √2b/a = b - (1 + b/a)t. |

Again, from

we can deduce that

(2) | 2bt²/a = b² - 2bt(1/2 + 2b/a) + t²(1/2 + 2b/a)² |

so that

(2') | t √2b/a = b - (1/2 + 2b/a)t. |

From (1') and (2'),

that is, a = 2b. From (1'), t = a/2 - 3t/2, or 5t = a.

**Note**: the steps from (1) to (1') and from (2) to (2') are based on unproven assumptions that the right hand sides in (1) and (2) are positive. If they are not, the derivations amount to claiming that

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### Conic Sections > Ellipse

- What Is Ellipse?

- Analog device simulation for drawing ellipses
- Angle Bisectors in Ellipse
- Angle Bisectors in Ellipse II
- Between Major and Minor Circles
- Brianchon in Ellipse
- Butterflies in Ellipse
- Concyclic Points of Two Ellipses with Orthogonal Axes
- Conic in Hexagon
- Conjugate Diameters in Ellipse
- Dynamic construction of ellipse and other curves
- Ellipse Between Two Circles
- Ellipse in Arbelos
- Ellipse Touching Sides of Triangle at Midpoints
- Euclidean Construction of Center of Ellipse
- Euclidean Construction of Tangent to Ellipse
- Focal Definition of Ellipse
- Focus and Directrix of Ellipse
- From Foci to a Tangent in Ellipse
- Gergonne in Ellipse
- Pascal in Ellipse
- La Hire's Theorem in Ellipse
- Maximum Perimeter Property of the Incircle
- Optical Property of Ellipse
- Parallel Chords in Ellipse
- Poncelet Porism in Ellipses
- Reflections in Ellipse
- Three Squares and Two Ellipses
- Three Tangents, Three Chords in Ellipse
- Van Schooten's Locus Problem
- Two Circles, Ellipse, and Parallel Lines

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