This problem was posted at Myōjōrinji temple in Gifu Prefecture in 1865. Here is its entry at the Sangaku Archive [coming soon].

In this problem, fix the outer radius of the fan and allow the inner radius to vary; tangencies are preserved by changing the fan’s central angle. Assume that this inner radius is adjusted so that the white circle radius is maximal. The goal is to show that, under this condition, the sum of the red and white circle diameters is equal to the outer fan radius.

For convenience, let the outer fan radius equal 1 and denote the inner fan radius [i]s[/i]. Let [math]\large w<g<r[/math] be the radii of the three other circles. It turns out that points A, B, and C lie along the same line; we will discuss the reasoning a bit later. For now, assuming this collinearity, we can observe that[br][center][math]\large [br]s+2r=s+2w+2g=1.[br][/math][/center][br]Further, by similar triangles, we have that[br][center][math]\large [br]\frac{g}{w}=\frac{s+2w+g}{s+w},[br][/math][/center][br]which can be rewritten in terms of only [i]s[/i] and [i]w[/i], using the first equation, as[br][center][math]\large [br]\frac{1-s-2w}{w}=\frac{1+s+2w}{s+w}.[br][/math][/center][br]This yields a quadratic in [i]w[/i], the positive root of which is[br][center][math]\large [br]w=\frac{1}{2}(\sqrt{s}-s).[br][/math][/center][br]

We wish to maximize [i]w[/i] with respect to [i]s[/i]. For someone who has learned calculus, the next step might be to take the derivative of [i]w[/i] and set it equal to zero. Thus,[br][center][math]\large [br]\frac{dw}{ds}=\frac{1}{2}\left(\frac{1}{2\sqrt{s}}-1\right)=0.[br][/math][/center][br]Solving this, we find [math]\large s=1/4[/math]. Hence, the maximal value for [i]w[/i] is [br][center][math]\large [br]w=\frac{1}{2}(\sqrt{1/4}-1/4)= \frac{1}{8}.[br][/math][/center][br]We can use the first equation again to find the radii of the other circles; they are[br][center][math]\large [br]g=\frac{1}{4} \text{ and } r=\frac{3}{8}.[br][/math][/center][br]In particular, we can see that the required equality holds:[br][center][math]\large [br]2w+2r=1.[br][/math][/center][br]

[center][icon]/images/ggb/toolbar/mode_parallelplane.png[/icon][/center]

There are two important ways in which a [i]wasanka[/i] would have done this problem differently. First, they would not have taken the derivative of a function in order to maximize it. This is a notion from differential calculus, which developed in 17th century Europe. Thus, like the Cartesian coordinate system, tools related to the derivative of a function would have been absent from traditional Japanese math due to the country's seclusion. [br][br]However, other tools exist for optimizing algebraic quantities. One such method, for quadratics, has been around for millennia. To get a quadratic expression for [i]w[/i], use the substitution [math]\large t=\sqrt{s}[/math]:[br][center][math]\large [br]w=-\frac{1}{2}t^2+ \frac{1}{2}t.[br][/math][/center][br]Notice that since [i]t[/i] increases with [i]s[/i], the quantity [i]w[/i] is maximized at corresponding values of the two. Now, consider the easily verifiable fact that[br][center][math]\large [br]-\frac{1}{2}t^2+ \frac{1}{2}t = \frac{1}{8} -\frac{1}{2}\left(t-\frac{1}{2}\right)^2.[br][/math][/center][br]Looking at the expression on the right, observe that the squared term is always non-negative. Therefore, the expression must be maximized when the quantity being subtracted from 1/8 is as small as possible, i.e., when [math]\large t=1/2[/math], thus [math]\large s=1/4[/math]. This approach is known today as [i]completing the square[/i], but dates back to Old Babylonian mathematics (c.~1700 BCE), and certainly was in the [i]wasan[/i] toolkit.

The second deviation we might make from a [i]wasanka[/i] relates to the fact that points A, B, and C are collinear. An easy way to prove this uses the theory of [i]inversive geometry[/i], which developed in Europe in the early to mid-19th century. Also known as [i]circle inversion[/i], this theory provides a precise framework for transforming the two-dimensional plane by "inverting it" with respect to a given circle. In essence, points inside that circle are mapped to points outside, and vice versa, according to some basic rules. These conditions force [i]any[/i] circle or line to map to another circle or line.

Consider a circle O with center A and radius [math]\large \sqrt{s} [/math]. Using results from inversive geometry, one can deduce that the given configuration of circles in the fan (regardless of whether the white circle is maximized) is only possible if the following pairs of objects are inverses of each other with respect to O: the inner and outer fan circles; the white and green circles; the red circle, with itself; and each radial side of the fan, with itself.[br]

Another theorem states that the centers of two circles that are inverses of each other must be collinear with the center of the circle of inversion. Hence, A, B, and C lie along a single line.[br][br]Would a [i]wasanka[/i] have argued this way? There is disagreement among scholars about the extent to which a theory of circle inversion developed in Edo Japan. Though a full-fledged theory is unlikely to have emerged, there is evidence that Japanese mathematicians (roughly contemporaneous with their European counterparts) established some of the [url=https://www.geogebra.org/m/vp4rbsjk#material/wvghuu4g]concepts underlying inversive geometry[/url]. As rigorous proof (in the tradition of Euclid) was not typically a requirement in [i]wasan[/i], these foundational ideas may have been sufficient to convince a [i]wasanka[/i] of the necessary collinearity.