## Sangaku: Find the Radii of the Inner Circles

Sangaku (算額) are Japanese geometric puzzles written on wooden tablets over 150 years ago. There have been several previous puzzles, but I didn’t see this one. Find the radii of the two inner circles in terms of x: Answer Let the radius of the smaller circle be r, and the larger be R. It is … Read more

## Japanese Temple Geometry

Hello, I was trying to solve this problem using descarte circle theorem for my maths report. I looked through the solution but I don’t understand the part in the answer, where it says the two solutions are $p_{n+1}, p_{n-1}.$ Can someone explain it for me. Thanks! Answer Consider this form of the equation 2(p_1^2 + … Read more

## Solving a sangaku circle problem using a system of equations

From the question “Sangaku Circle Geometry Problem”: Given a and b, find c. (The enclosing circular segment is not necessarily a semicircle.) The answer is c = \frac{a(\sqrt{a} + \sqrt{b})^2}{4(3b + \sqrt{ab} )} I’m curious how one would solve the given puzzle using a system of equations. My first attempt was to describe all circles … Read more

## Equilateral triangle and very peculiar inscribed tangent circles

The problem is to find the length of the size of the equilateral triangle below I found one equation: Let R be the radius of the big circle whose red arc touches the two purple circles. Let A be the triangle vertex on which the red circle touches the green side. Let’s call B the … Read more

## Japanese Temple Geometry Problem: Two tangent lines and three tangent circles.

I am working on my Senior Thesis for my Bachelor’s Degree in Mathematics. My project involves Japanese San Gaku problems, and moving said problems from Euclidean Geometry to Spherical and Hyperbolic Geometry. I’ve been working on a particular problem for weeks now. The problem is stated as follows: Problem 1.2.5: A circle $O(r)$ has its … Read more

## Is there a way to reduce a specific quintic to cubic?

A polynomial in two variables, t and c, is quintic in t and quartic in c: 16t5−8c(5c+2)t4+c2(25c2+20c+36)t3−4c(11c3+8c2+5c+2)t2+8c2(3c2+3c+2)t−4c3(c+2), (25t3−44t2+24t−4)c4+(20t3−32t2+24t−8)c3−4t(10t3−9t2+5t−4)c2−8t2(2t2+1)c+16t5. It is related to An ancient Japanese geometry problem Given isosceles △ABC with legs |AC|=|BC|=1 and base |AB|=c, find t=|BD| which provides the same radius for all three circles, inscribed in triangles ADC, ABE and BDE. The … Read more

## Sangaku Circle Geometry Problem

I’m having difficulties with this Sangaku problem and was hoping for some help! Five circles (1 of radius c, 2 of radius b, and 2 of radius a) are inscribed in a segment of a larger circle (note: this segment does not have to be a semi-circle). Given a and b, find c. For example, … Read more

## Exploring a Sangaku problem: proving a dilated circle is circumcircle

ΔABC is an equilateral triangle with D being the midpoint of BC. ΔDEF is also an equilateral triangle such that E,F are on minor arc BC of the circumcircle of ΔABC with DE∥AB & DF∥AC. ΔBDG is equilateral such that E lies on DG.  Let H be the point on the circumcircle of ΔDEF such that HF is the diameter. Prove that H is the circumcenter of ΔBGF. The original Sangaku problem is to find the ratio of the sides of triangles ABC and DEF. It is not … Read more

## How these circles are congruent?

Here is a problem involving curvilinear incircles and mixtilinear incircles. Let a triangle△ABC have circumcircle γ.It’s A-Excircle tangency point at sideBC is D Let γ1 be the circle tangent to AD,BD,γ also γ2 is the circle tangent to AD,CD,andγ.prove that γ1 and γ2 are congruent. I have tried to prove the converse of the problem. … Read more

## Sangaku: to prove one of the intangents is parallel to BCBC

Given an acute triangle △ABC whose incircle is I(r). Let O(R) be the circle through B and C and which touches I(r) interiorly. Show that the circle P(p) which is tangent to AB, AC and O(R) (externaly) is such that one intagent line from P(p) and I(r) is parallel to side BC. I have seen … Read more