This problem was posted at the Gion (now Yasaka) shrine in Kyoto in 1749. It is located on the Sangaku Archive at [URL].
In the circular segment below, the chord has length [i]a[/i] and its perpendicular bisector has length [i]m[/i]. Consider the square of side [i]s[/i] and circle of diameter [i]d[/i] as shown. Given the quantities[br][center][math]p=a+m+s+d[/math][/center]and[br][center] [math]q=\frac{m}{a}+\frac{d}{m}+\frac{s}{d}[/math][/center]find [i]a [/i]in terms of [i]p [/i]and [i]q[/i].
The values of [i]p [/i]and [i]q[/i] depend on geometrical aspects of figure. You can try adjusting the radius of the big circle and moving the red triangle point in the middle of the chord to see how they change [i]p[/i] and [i]q[/i]. [br][br]Notice also that the value of [i]a[/i] varies with [i]p[/i] and [i]q, [/i]but in a non-obvious way. In 1774, mathematician Ajima Naonobu presented this relationship as a tenth degree polynomial in [i]a[/i] with coefficients in terms of [i]p[/i] and [i]q[/i]. This polynomial sufficed as a solution, since any specific values given for [i]p[/i] and [i]q[/i] would yield a polynomial in the variable [i]a[/i] that could be solved numerically; however, Ajima did not address the issue of choosing which of the potentially ten distinct roots would be the actual length of the chord. As of this writing, no one has found a polynomial solution with lower degree.[br][br]Ajima's solution is also notable for its use of a 4x4 determinant calculation using a form of cofactor expansion, which Japanese mathematicians developed independently from their counterparts in Europe. For all its algebraic challenges, however, the Gion shrine problem is fairly simple geometrically: there are three right triangles--with sides composed of the five basic variable lengths--that completely describe the construction. You can display these above.
This problem is algebraically very difficult as stated. For a more approachable version, change the two given quantities to simpler ones and use numerical values. For example, "given that [math]m=10[/math] and [math]d=9[/math], find [i]a[/i] and [i]s[/i]."