Project 5: Same polygons with different circles and a surprise (GER, RO, IT, CRO)
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Continuations:
German students will talk about your work next week and give some comments in the Forum!
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.
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.
Carnot's Theorem and Japanese Proof
Explaining through Carnot’s Theorem the Japanese Theorem.by Norma Lisa Neiman
Germany had the idea for a project (grade 5, 6) but we need a tool before starting
A Romanian student created a tool for the inner circle and uploaded it.
German students worked on pentagons (see worksheet) and made a surprising mathematical discovery. A common artwork was created, too.
German students asked older students in a video for a proof.
Croatian students constructed inner circles on paper and made the same discovery.
Italian students of grade 10 started with a proof of the theorem.
A talented German started and a trainee teacher and some teachers of the project starte with the proof, too.
We also looked on the internet for a proof. Up to now not found ;-)
Italian students of grade 9/10 started with a proof for triangles (two parts)in a video.
A visual proof done by the Italian teacher.
Italian students of grade 9/10 and teachers proved the Japanese Theorem through the Carnot's Theorem. Third Part GeoGebra file and through iteration.
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)
For interested teachers and older students: International conversation in Twitter started by a Trainee teacher at our School. There are two Proofs.
Monika, We can't wait to see the result! Norma Lisa Neiman