Concurrence

Our main topic for this 2-week stretch will be concurrence--that's when three lines meet at a single point (which is pretty rare).

On Tuesday April 14th, we'll be investigating some concurrence situations. To have the most fun, you will need your patty paper (little square tracing paper pieces), a pencil and a straight edge.

Lemmas (approximately April 14)

Before Tuesday, you can start working on this assignment to prove some Lemmas: a lemma is a small helper-theorem that you use as a stepping stone to get to something more interesting.

Here's the assignment. It consists of 5 lemmas for you to prove. The statements are on the first page, some written hints are on the second page, and I have walkthrough videos for you to follow along for anything you couldn't prove without hints (or if you just want to check that you're doing it right).

Write it up on pencil and paper, and photograph it to get it into a Word document to turn in. I haven't made a dropbox for it yet, but I will soon!

Video of Lemma 1
Video of Lemma 2
Video of Lemma 3
Video of Lemma 4a
Video of Lemma 4b

Exploring and naming triangle centers (Approx April 16)

If you weren't able to make it to the Zoom meetings on April 14th and 16th, please go watch those in Canvas. I have edited them down, to do my best to make them watchable, and they have lots of important content that I haven't put out anywhere else yet.

I have two Geogebra explorations for you to do (assignments) in the class Geogebra Group:

1. Perpendicular Bisectors of Sides and the Circumcenter
2. Angle Bisectors and the Incenter.

I also have a project-part for you to get started on that I explained in the April 16th discussion.

Instructions for Triangle Center Project (part 1)

Create two triangle center map puzzles (using two different centers from the choices: incenter, circumcenter, centroid, orthocenter). What you need to turn in is two puzzles of the type:

Where is the [incenter/circumcenter/centroid/orthocenter] of the triangle with vertices at [3 map locations]

For example:
Where is the centroid of the triangle with vertices at York, Manchester and Skipton in the UK?

Note 1: the triangle center should be visible on a Google Maps map with the same zoom as the three map locations, and all four locations should be named locations. For example, the answer to the example question is Bradford which is another city name.

Note 2: the locations do not have to be cities, but they should be places whose names are labelled in Google Maps, and you need to give enough information that someone else could Google the location and find it.

Skills videos (watch these if you have trouble with the project):

Putting a Google Maps map screenshot into Geogebra
Hiding and showing in Geogebra
Creating a dragable incenter in Geogebra
Creating a dragable circumcenter in Geogebra
Creating a dragable centroid in Geogebra
Creating a dragable orthocenter in Geogebra

Proving concurrence: (on or about April 21)

Prove that all four poiints of concurrence really exist.

Assignment: Complete all of these proofs.

Video of me explaining explaining key things to notice in the assignment.

Assignment (Euler Line): Investigate the "Euler line" in the Geogebra group task.

Triangle center project part 2 coming soon

Part 2 is a quiz (?) type assignment that has 5 questions: the first 4 are finding triangle centers on maps (two of which questions came from ones that you all submitted!). The last question asks you a related question about triangle centers, and is intended to assess your understanding of what the important properties are of the four types of triangle centers.

The file with the quiz questions, along with files with map images are in the appropriate assignment in Canvas (open the assignment, and the files are in the assignment description.

Email if you have questions!