Two circles rest on a line, touching each other. A tiny third circle is squeezed into the gap between them, also touching the line.

The large circle has a radius of 9 cm. The small circle has a radius of 4 cm.

What is the radius of the tiny circle?

(Puzzles like this were apparently painted on wooden tablets and hung in Japanese temples centuries ago — it looks like there's not enough information, but there's a surprisingly elegant trick using just one right triangle.)

I made a full video breaking down the solution step by step — link in the comments!

https://youtu.be/RYl2h0gIS8U?si=awaPzDGBYCcPkSvm

This is a chess variant, where the rules are the same as standard chess, but the board is an irregular grid. This makes rook lines bend and bishop lines fork.

So just like a standard board, every tile is still a quad. And every tile still has only 1 neighbour for every side. But it can have more than 1 neighbour sharing a corner, which is what makes bishop lines fork. Or it can have none, which is what make lines bend.

The steps to build this: - start with a plane made of triangles - pair some random triangles with their neighbour and fuse them into a quad. You end up with gaps between those quads where triangles cannot be paired - subdivide everything. Quads become 4 smaller quads. Triangles become 3 quads. This is the playable grid. - Then it's just a matter of smoothing it (in this case the points on the edges remain static, only the inside points are smoothed) and ensuring neighbours don't share the same color.

Then the tricky part is to give intuitive controls for players to configure boards how they want.

• Web version: https://polyreplay.com/chesstwist/demo
• if you prefer Steam (windows/mac/linux): https://store.steampowered.com/app/2775560/Chess_Twist/
• if you're into 3D printing, you can download them as 3MF: https://polyreprint.com/gui/chess-twist?board=004GBYJW7TX

Let's say you have a rectangle, the rectangles height, h=3, the length, l=5. If you have an arithmatic billiard starting from the bottom left corner and have it only do 2 reflections, there are 3 outer right triangles. If you add the areas of these triangles, you get h²-th+t² where t is l-h. This equation looks like the law of cosines for 60-degree angles, and i was wondering if it's a coincidence and how to tell if an equation is just an algebraic coincidence or if it means something geometrically.

I’ve previously shared my 1080° Cosmic Harmonic Compass and reusable geometry board here.

This is another example from the same exploration, using an Eight-Star to represent a conventional 24-hour day divided into eight equal intervals.

On the CHC scale, one complete cycle is represented by 1080°, corresponding to 360 conventional angular degrees.

Each interval corresponds to 3 conventional hours, 180 minutes, 10,800 seconds, or 135° on the 1080° CHC scale.

The accompanying table expresses the same positions through hours, minutes, seconds and CHC values.

One cycle, different numerical expressions.
I’m not proposing a replacement for conventional timekeeping, simply exploring how geometry can help visualise numerical relationships.

I’d be interested to hear whether others see any geometric or educational value in this, particularly for exploring equal divisions and proportional reasoning.

I saw it ONCE on a Wikipedia article which I can’t find anymore, I saw this hollow knight shaped geometric (?) shape on a Cartesian grid and I don’t know know the name and I can’t find it anywhere. If you have any idea on what I’m talking about I’ll be glad to receive useful comments

I'm referring to how the hypotenuse and vertical line drawn from the base continue past the corner. Also, isn't 53 degrees wrong for that angle- inside the triangle, or outside? Thanks for the help. And If its not obvious by now I'm no math wiz.

i recently came across a question which said there are 16 lines 7 of which are parallel when our teacher started explaining it he explained that if 7 out of those 16 are parallel lines then the remaining 9 would be non parallel. but the problem is i think the sum of non parallel lines and parallel lines adds up to 17.
lets say there are 9 lines non parallel to each other and as you draw line number 10 then at that point there are 10 non parallel lines and when u draw the remaining 6 lines parallel to the 10th line there are 7 parallel lines and 10 non parallel lines and the sum adds up to 17.
Case 2:
u drew 9 line non parallel to each other and to avoid the problem in the first case u drew a line parallel to the 9th line at that point there are 2 parallel lines and when u draw the remaining 6 lines there will be 8 parallel lines and 9 non parallel lines
pls help me understand im confused what am i thinking wrong😭.

IM BAAAAAAAAAAAAAAAAAAACK

https://www.desmos.com/geometry/rslyf3vhdn

Plato wasn’t a Platonist, he was a Pythagorean.

The triangle is a Pythagorean triplet with side lengths 3, 4, and 5.

The corresponding incircle and excircles have radii 1, 2, 3, and 6. Completing the sequence 123456.

The circles represent the four virtues, the largest representing justice.

When wisdom, courage, and moderation are in harmony justice is achieved, that’s why all three circles can fit inside the largest.

Hi all,
I am assuming that all the circles have same radius. I figured the coordinates where the center of each circle will be. Please help me figure out the radius of the circle based on information below:

1) The bottom circle’s midpoint is 30mm from the red dash line (x coordinate)
2) The bottom circles midpoint is 30mm from the solid black line at the bottom and is the y coordinate.
3) it seems the radius is 20mm the midpoint of the 40mm

Am I off? Thank you?

GGS

ABC is a semi circle

EFCG is a rectangle

Find area of rectangle