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[Geometry] Sangaku Three Circles Problem

2021-07-03 14:12 作者:AoiSTZ23 0人讀過 | 我要投稿

By: Tao Steven Zheng (鄭濤)

【Problem】

Figure 1 shows a famous Japanese sangaku (算額) problem from Gunma prefecture in 1824. Suppose the radius of the left circle is $%7Br%7D_%7B1%7D$ , the radius of the right circle is $%7Br%7D_%7B2%7D$ , and the radius of the circle at the center is? $%7Br%7D_%7B3%7D$ , determine the formula for the radius of center circle.

【Solution】

Let? $%7Br%7D_%7B1%7D%2C%20%7Br%7D_%7B2%7D%2C%20%7Br%7D_%7B3%7D$ ?represent the radius of the left circle, right circle, and center circle respectively. According to Figure 2,? $AG%3D%7Br%7D_%7B1%7D$ ,? $BH%3D%7Br%7D_%7B2%7D$ and $OC%3D%7Br%7D_%7B3%7D$ . Furthermore, $AE%3D%7Br%7D_%7B1%7D-%7Br%7D_%7B3%7D$ , $BF%3D%7Br%7D_%7B2%7D-%7Br%7D_%7B3%7D$ , $BD%3D%7Br%7D_%7B2%7D-%7Br%7D_%7B1%7D$ ?and $AB%3D%7Br%7D_%7B1%7D%2B%7Br%7D_%7B2%7D$ . Let $EO%3Dx$ and $FO%3Dy$ , then $AD%3Dx%2By$ .

?

Applying the Pythagorean theorem, we get

$x%5E2%2B(%7Br%7D_%7B1%7D-%7Br%7D_%7B3%7D)%5E2%3D(%7Br%7D_%7B1%7D%2B%7Br%7D_%7B3%7D)%5E2$

$y%5E2%2B(%7Br%7D_%7B2%7D-%7Br%7D_%7B3%7D)%5E2%3D(%7Br%7D_%7B2%7D%2B%7Br%7D_%7B3%7D)%5E2%0A$

$(x%2By)%5E2%2B(%7Br%7D_%7B2%7D-%7Br%7D_%7B1%7D)%5E2%3D(%7Br%7D_%7B1%7D%2B%7Br%7D_%7B2%7D)%5E2%0A$

Simplify the above three equations as

$x%5E2%3D4%7Br%7D_%7B1%7D%20%7Br%7D_%7B3%7D%0A%0A$

$y%5E2%3D4%7Br%7D_%7B2%7D%20%7Br%7D_%7B3%7D%0A%0A$

$(x%2By)%5E2%3D4%7Br%7D_%7B1%7D%20%7Br%7D_%7B2%7D%0A$

Taking the positive square roots give

$x%3D2%5Csqrt%7B%7Br%7D_%7B1%7D%20%7Br%7D_%7B3%7D%7D%0A%0A$

$y%3D2%5Csqrt%7B%7Br%7D_%7B2%7D%20%7Br%7D_%7B3%7D%7D%0A%0A$

$x%2By%3D2%5Csqrt%7B%7Br%7D_%7B1%7D%20%7Br%7D_%7B2%7D%7D%0A$

Hence,

$%5Csqrt%7B%7Br%7D_%7B1%7D%20%7Br%7D_%7B3%7D%7D%2B%5Csqrt%7B%7Br%7D_%7B2%7D%20%7Br%7D_%7B3%7D%7D%3D%5Csqrt%7B%7Br%7D_%7B1%7D%20%7Br%7D_%7B2%7D%7D%0A%0A$

$%5Csqrt%7B%7Br%7D_%7B3%7D%7D%20%20(%5Csqrt%7B%7Br%7D_%7B1%7D%7D%2B%5Csqrt%7B%7Br%7D_%7B2%7D%7D)%3D%5Csqrt%7B%7Br%7D_%7B1%7D%20%7Br%7D_%7B2%7D%7D%0A%0A$

$%5Csqrt%7B%7Br%7D_%7B3%7D%7D%3D%5Cfrac%7B%5Csqrt%7B%7Br%7D_%7B1%7D%20%7Br%7D_%7B2%7D%7D%7D%7B%5Csqrt%7B%7Br%7D_%7B1%7D%7D%2B%5Csqrt%7B%7Br%7D_%7B2%7D%7D%7D%0A$