## Project 5

• Project 5: Same polygons with different circles and a surprise (GER, RO, IT, CRO)

• Worksheet  and creative, common results (GER)
• Looking for help from a student (GER) -> see here .
• Student Sophia from Romania has sent it. Thank you! So we can now work on this project!

You can change the settings here to get english subtitles!

Continuations:

• Finding the number of different carvings for pentagons, hexagons, ...
• How many arrangements of triangles in a pentagon, hexagon, ... ?
• Is there a formula for the number of different carvings?
• Proof for pentagons, hexagons, ...
• The circles could be animated at the end e.g. by rotation

• Construction on paper (CRO)

• Proof of Japanese Theorem from Italian students first part:

A triangle is always cyclic. Let's prove the Japanese theorem, dividing it into parts. Let's start from "One and only one circle passes through three given non-collinear points" . and that these three points are even the vertices of a unique triangle. Let's explore some of the properties: the incenter and the incircle of the triangle and the inradius and the area.

• Second Part:

The circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. Let's explore the relation between the area of ABC the inradius and circumcenter.

• Third Part

Carnot's Theorem and Japanese Proof

• Proof of  the Japanese Theorem through iteration by Norma Lisa Neiman

Carnot's Theorem

Explaining through Carnot’s Theorem the Japanese Theorem.by Norma Lisa Neiman

• #### Cooperation and communication in this project

##### History 1

Germany had the idea for a project (grade 5, 6) but we need a tool before starting

##### History 2

A Romanian student created a tool for the inner circle and uploaded it.

##### History 3

German students worked on pentagons (see worksheet) and made a surprising mathematical discovery. A common artwork was created, too.

##### History 4

German students asked older students in a video for a proof.

##### History 5

Croatian students constructed inner circles on paper and made the same discovery.

##### History 6

Italian students of grade 10 started with a proof of the theorem.

##### History 7

A talented German started and a trainee teacher and some teachers of the project starte with the proof, too.

##### History 8

We also looked on the internet for a proof. Up to now not found ;-)

##### History 9

Italian students of grade 9/10 started with a proof for triangles (two parts)in a video.

##### History 10:

A visual proof done by the Italian teacher.

##### History 11

Italian students of grade 9/10 and teachers proved the Japanese Theorem through the Carnot's Theorem. Third Part GeoGebra file and through iteration.

##### History 11

A lot of people (in Europe)- mostly students and a prof at university have worked on this "simple" problem. A trainee teacher in my school has started a tweed on twitter and a lot of people answered. He also made a proof (3 pages) and it will take some time to prepare a presentatation with all twitter answers and his own proof. We can be very curious of it! Monika (GER)

##### History 12

For interested teachers and older students: International conversation in Twitter started by a Trainee teacher at our School. There are two Proofs.

##### History 11

Monika, We can't wait to see the result! Norma Lisa Neiman