【CK's Geometry Talks】 Page

Created May 30, 2023

The book by Hidetoshi Fukagawa and Dan Pedoe, Japanse Temple Geometry Problems (San Gaku) (below left), was the first book in which many problems were collected and discussed. Later, Hidetoshi Fukagawa and Tony Rothman published another comprehensive book, Sacred Mathematics: Japanese Temple Geometry (below right), in which many problems were explained along with a detailed history account.

In the 2023 Spring months, while browsing YouTube for something for my own research, a few videos discussing the Japanese Temple Geometry problems were found. Because I knew the topics, I watched nearly all of them with great interests. However, I found that some videos did discuss one or two Japanese Temple Geometry problems, while some others failed to address the problem properly. Moreover, some even incorrectly identified the Japanese Temple Geometry problems. Because I have the two definitive books and read many related papers, I believe I could do a better job in explaning these Japanese Temple Geometry problems with rigor. This was my initial thought of making my own videos. Then, in the planning stage I realized that I could do more by

• grouping similar problems together so that the viewers do not have to watch multiple isolated videos;
• discussing interesting and beautiful theorems in elementary geometry such as the Morley Theorem (i.e., the adjacent trisectors of a triangle meet at an equilateral triangle), the Feuerbach Theorem (i.e., the nine-point circle of a triangle is tangent to the incircle internally and the three excircles externally), the Steiner-Lehmus Theorem (i.e., if two angle bisectors of a triangle have equal length, the triangle is an isoceles) an many more.
• presenting interesting topics in elementary geometry such as coaxial circles, inversion, symmedians, harmonic range, cross ratio, etc.
• reviewing the impossibility of trisecting an arbitrary angle and doubling a cube, the constructibility of a regular 17-gon, the unsolvability of rational coefficient polynomials of degree 5 and higher using +, -, *, / and radical operators, and so on.

There are many possible topics and the above only named a few.

Please note that these videos are purely work based on my personal interests. I did not receive anything from YouTube and elsewhere. A big "like" 👍 is very much appreciated.

Thank you!

Table of Content

No.	Content	Videos	Slides	Notes	Date
1	Given two circles of equal radius that are tangent to each other externally, if they are both tangent to the hypotenuse of a right triangle and each is tangent to one of the other sides, find the radius in terms of the side lengths: In this lecture, the existence and uniqueness of this pair of circles are also proven.	A Simple Japanese-Temple-Geometry- Problems like Problem	Slides	Lecture Notes	April 8, 2023
2	What if the two equal radius circles are tangent one of the two non-hypotenuse sides? Find the radius in terms of the side lengths: What if the given triangle is an arbitrary triangle? In this case, the two equal radius circles are tangent to one of the three sides and each of the circle is tangent to one of the other sides. In this lecture,we solve these two problems and the existence and uniqueness of this pair of circles are also proven.	A Variation and A Generalization of the Problem in EP1	Slides		April 19, 2023
3	This lecture focuses on two seemingly unrelated sets of problems and shows that they are actually very closely related. The first set of problems has two circles that are tangent to each other and tangent to a fixed line. What is the radius of the circle, say C1, that is tangent to the two given circles and the given line? Let one of the two given circles be A. Then, circle A and circle C1 determine another circle C2 in the same way. What is the radius of this circle C2? This process is able to produce an infinite number of circles C1, C2, C3, ..., Cn. What is the radius of circle Cn in terms of the two given circles? The first set of problems addresses this question. Then, this lecture proceeds to the second set of problems. Given a circle and a chord, from the largest circle that is tangent to the given circle and the chord one can construct an infinite number of circles that are tangent to each other and also tangent to the given circle and chord. Is there a relation among the centers of these circles? Answer: these centers are on a parabola! These two sets of problems seem to be unrelated. However, the second set actually serves as a kind of the converse of the first. More precisely, given a line and two fixed points on the line, each pair of circles that are tangent to each other and tangent to the line at the two fixed points uniquely determine a third circle that is tangent to the give circles and the given line. There are an indinite number of such circles. In this section, we will show that the centers of these circles lie on a parabola and all circles are tangent to a common circle. Conversely, given a fixed circle and a chord satisfying a mild condition, then any circle that is tangent to the given circle and the given line uniquely determines a pair of circles that are tangent to each other and also tangent to the give chord at the intersection points of the chord and the the given circle.	Seven Japanese Temple Geometry Problems	Slides	Lecture Notes	July 23, 2023
4	This is the first talk on the Pythagorean Theorem and the Pythagorean Identity. Elisha Scott Loomis made an incorrect claim in his well-kown book, "Pythagorean Proposition' published in 1907, that the Pythagorean Theorem cannot be proved by trigonometric techniques because the fundamental identity (i.e., the Pythagorean Identify in today's terminology) depends on the Pythagorean Theorem. Therefore, proving the Pythagorean Theorem using trigonometry could be in a circular reasoning. This claim/reasoning is WRONG! This is the first lecture of a 3-episode series. We shall prove that the Pythagorean Identity can be proved without using the Pythagorean Theorem. In fact, we shall show that the angle difference and angle sum identities are independent of the Pytagorean Theorem and the Pythagorean Identity. Then, the angle difference identity implies the Pythagorean Identity immediately. We also prove that the derivative computation of sin(x) and cos(x) are also independent of the Pythagorean Theorem and the Pythagorean identiy, and show that the double angle identities, which are direct consequences of the angle sum identities, implies the Pythagorean Identity. Consequently, we proved that Loomis was wrong! The next two lectures will focus on possible new proving techniques and the proof by Jaskson and Johnson presented in early 2023. Why is Jackson-Johnson proof important? This is because their proof was claimed to be an impossible proof as their used trigonometry. We will show in the third episode of this series that the use of trigonometry in the proof of Jackson and Johnson can be eliminated completely with the new technique.	The Pythagorean Theorem I	Slides	Lecture Notes	October 31, 2023
5	This is the second talk on the Pythagorean Theorem and the Pythagorean Identity. This lecture discusses a new method of computing the length of a line segment and the area of a shape based on similarity. With the help of this new method, we are able to re-prove some proofs discussed in Lommis book. Additionally, by using the Lemoine/Grebe/Symmedian point and one of its many interesting properties we are able to provide a possibly new proof. The next (and actually last) lecture, we will discuss the proof by Jackson and Johnson.	The Pythagorean Theorem II	Slides		November 30, 2023
6	This is the third talk on the Pythagorean Theorem and the Pythagorean Identity. This lecture discusses a pure geometric proof based on the 2023 proof of Jackson and Johnson. Note that the proof of Jackson and Johnson used trigonometry, which according to Lommis is impossible. After presenting this pure geometric prrof, the original trigonomatry based proof is discussed. Therefore, the trigonometric component used in Jackson-Johnson's proof can easily be eliminated. This is the last lecture of this 3-lecture series. Hope you like it and these 3 lectures are helpful.	The Pythagorean Theorem III	Slides		December 2, 2023

Update History

1. Created: May 30, 2023. The first two lectures are online.
2. Update #1: August 2, 2023. Episode 3 is added.
3. Update #2: November 6, 2023. Episode 4 (the first episode of The Pythagorean Theorem series) is added.
4. Update #3: November 30, 2023. Episode 5 (the second episode of The Pythagorean Theorem series) is added.
5. Update #4: December 2, 2023. Episode 6 (the third episode of The Pythagorean Theorem series) is added.
6. Update #5: January 16, 2024. Three long articles are added, each of which corresponds to its related lectures. Please see the Notes column.