Geometry

1) Now that we've got the similar triangle result from (: Josh Random Triangle letter , followed by Jim Tanton's essay on breaking a stick into three parts to create a triangle ) to work with here's a classic geometry theorem whose proof involves similar triangles (it's one of several cases collectively called the "Power of a Point" Theorem). Let A be a point inside a circle and draw any two lines through A cutting the circle at points B, C, D and E (see figure below). Using similar triangles show that |AB| x |AE| = |AC| x |AD|. Hint - find two similar triangles by adding a few chords (and show the angle equality you need by thinking about a few more chords). Note that here we're using a fairly standard notation |AB| to stand for the length of the line segment AB (to distinguish between the actual line segment itself, and its length - often texts will write AB with a bar over top to mark the segment itself, and then reserve AB to stand for its actual length, but other texts use AB to stand for both).

2) Continuing this theme, prove that this also holds when the point is outside of the circle, i.e. that |AB| x |AE| = |AC| x |AD| in the following diagram as well (it's almost the same proof as the one in the previous problem - you just need to find two similar triangles involved for this set-up (again draw in a few more chords, and then track angles around to show that you're actually dealing with similar triangles):

3) And here's one more problem that's quite tricky using similar triangles... (don't despair if this one eludes you - it's really quite a challenge!) First, let ABC be any triangle, and let D be the point on segment BC so that AD is the angle bisector of the angle at vertex A (i.e. the two angles marked "alpha" are equal in the figure below).

Now prove that the ratio of the lengths of the triangle sides AB to AC is equal to the ratio of the lengths of segment BD to DC (i.e. |AB| / |AC| is equal to |BD| / |DC|). This is a pretty challenging problem, and so here's a relatively big hint as to how to get started with this problem if you need it - but note that there are several ways to prove this, so you might find something quite different from this. 4) And a last similar triangle problem for you - this was shared with the rest of the teachers in a Capstone course recently - it's tricky but doable if you look for as many similar triangle pairs as possible - and as you go through the algebra you'll notice that one of the variables essentially disappears- good luck! You'll need to work with the lengths of CE and EB along the way - call them X and Y to start with! You are asked to find the length of AD given the other two lengths that are marked, along with the fact that DE is parallel to AB, and FE is parallel to DB.

5) Pythagorean Proof Party time! For this question I'd like you to find your favorite Pythagorean proof and please be prepared to share this favorite Pythagorean proof. Please find an image or some other electronic file/video that you can share with others please type up a short description of your chosen proof highlighting why you think it's special try to convince the others of your choice as being the best/most elegant/coolest/etc. 6) As a last part of your Pythagorean Theorem sleuthing find out which U.S. president came up with a proof of the Pythagorean Theorem, and under what circumstances. 7) Here's a bit more about the classic book by E.A.Abbott written over a century ago titled "Flatland" - it's all about what life would be like if we lived in two dimensions instead of in three please write a comment about the aspect of Flatland that you considered most surprising or peculiar - or just a note about the social aspects of the Flatland society! 8) Continuing with the idea of passing a 3D object through Flatland (i.e. through a plane), consider what would happen if a pyramid (such as one of the classic Egyptian square based pyramids) passed through a two dimensional plane. The figures that would appear would vary depending on the orientation of the pyramid as it's being passed through the plane. Draw a snapshot sequence of three different possible pass-throughs (similar to the three different pass-on cube - see the cubes passing through Flatland (Links to an external site.) Links to an external site. website for inspiration). Each of your snapshot sequences

should show at least five images so that it's clear that you've figured out how the pyramid would look all the way through the pass through.