Chapter 1: Introduction to Sangaku

This article examines a classic Sangaku problem, which originates from Japan and showcases intricate geometric relationships. The specific problem we will explore is depicted on a tablet displayed at the Tokuraji Temple in Miyagi Prefecture, dating back to 1909.

In this intriguing scenario, we encounter two congruent circles that intersect, each passing through the center of the other. A blue circle is inscribed to touch both of these circles internally, while two smaller red circles fit snugly in the upper and lower gaps created by the configuration. The challenge lies in determining the relationship between the sizes of the blue and red circles.

Tip: Remember, Pythagorean principles can be quite beneficial in solving this!

Pause for a moment, grab a pen and paper, and give the puzzle a shot before proceeding to the solution!

Section 1.1: Analyzing the Configuration

To tackle this problem, it's effective to concentrate on one aspect of the diagram. By annotating the elements, we can extract essential information for our calculations.

Let's denote the radius of the red circle as 1. For the blue circle, we'll use the variable ( r ) for its radius. The line segment AD connects the centers of the two overlapping yellow circles and is equal to the radius of these yellow circles, which coincidentally is also the diameter of the blue circle.

Thus, we derive that AE measures ( 2r ), leading us to calculate AC as follows:

[

AC = AE - CE = 2r - 1

]

At this point, we observe that triangle ABC forms a right triangle, allowing us to apply the Pythagorean theorem.

Section 1.2: Applying the Pythagorean Theorem

Through a series of algebraic manipulations, we arrive at the following equation:

Ultimately, we find that when the radius of the red circle is set to 1, the radius of the blue circle equals 3!

This discovery reveals that the area of the blue circle is nine times greater than that of the red circle!

Isn't that fascinating?

What was your approach to this puzzle? Feel free to share your thoughts in the comments below; I'm eager to hear from you!

Chapter 2: Additional Resources and Challenges

For those interested in delving deeper into similar mathematical challenges, I recommend checking out the following videos:

The first video titled A Japanese Temple Problem, a Sangaku provides an in-depth exploration of this geometric conundrum.

The second video, A Sangaku Puzzle (Japanese Temple Problem), further elaborates on these intriguing puzzles and offers additional insights.

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Happy solving!