Two Boxes in a Circle
The March MathsJam had an interesting geometry puzzle.
geometry
15-75-90 on the quarter circle
I just saw a nice visual proof of the ratios for the 15-75-90 on the internet via @ilarrosac. This one works via symmetry and the Pythagorean theorem.
Two Circumcircles Walkthrough
I really enjoyed working through this problem from Stanley Rabinowitz I saw recently on Twitter. This was one of those problems where I circled around it a...
Cyclic Quad Area Problem
What follows is a walkthrough of the problem above which I had a lot of fun playing around with. Most of the solutions I saw online used trigonometry and so…
More Heron’s Formula Extensions
Continuing on the theme from the last post: going-one-step-beyond-herons-formula here’s another problem showing the power of the full conceptual framework...
Going one step beyond Heron’s Formula
Above I’ve found the incenter of the triangle at the intersection of the angle bisectors and rewritten all segments lengths in terms of the semi-perimeter a...
Pershan Geometry Walkthrough #2
This is the continuation of a series from First Post I’m following Michael along and capturing my problem solving process for comparison. To capture the pro...
Everything is Connected (yet again)
The other week I was checking out the geometric puzzles at https://sciencevsmagic.net/geo/ as part of a MathsJam evening. A small part of the process requir...
Walkthrough of Geometry Problem from Michael Pershan
This post is motivated by a conversation I had with Michael where I asked if he would be willing to document his problem solving process and if so I would d...
Dot in a Box: Addendum
This is a continuation to my last post: dot-in-a-box
Dot in a Box
[@fleonsotelo]
Another example of a problem that’s much harder one way
We had a snow day this Monday and so there was no Math Club. Instead, I’ve written a continuation in my series of posts on the curious way geometry problems ...
Cool walk through for geometric transforms
[Since its AMC 8 today - here’s a geometry walkthrough instead for the week]
15-75-90 Alternate Forms
Its been a while since I’ve talked about one of my favorite triangles the 15-75-90. So here’s a short post on a new detail that I realized about them the oth...
Angle Pairs
Visual proof that $ \angle{BCA} + \angle{DCE} = \pi / 4 $ or alternatively $ \arctan \left( \dfrac{u-v}{u+v} \right) + \arctan \left(\dfrac{v}{u}\right) = \p...
Regular Heptagon puzzle walk through
There are too many wildly different an interesting ways to attack this problem to not document.
Sometimes one way is trickier than the other: angle bisectors
Here’s another example of one the most interesting parts of geometry for me. (courtesy of a mathjam participant last month) See: earlier post Given an isosc...
Pentagon + Quadrature of the Parabola
Last week, I saw this really fun parabola problem from @diegorattaggi and I became interested for two reasons:
Langley’s Adventitious Angles
I’ve been working through “Geometry Revisited” and have come to a section of old chestnuts one of which was Langley’s Adventitous Angles.
Noticing and Wondering is for Everyone
One very common prompt seen online is “What do you notice and wonder?” I like the frame of mind it suggests and often use it or variants with the kids in Ma...
Euler Line Walkthrough
http://www.gogeometry.com/problem/p742-circumradius-orthocenter-centroid-midpoint-distance-square.htm
Trigonometry Walk Through Three Ways
I saw the following trigonometry problem the other day and decided it would make another good walk through since it hits several themes I’ve been exploring.
Spring Break - Geometry Walk Through
Note: with Spring I really have no kids. Even my own are with my parents so here’s an old walk through I had laying around. On reread after a significant ga...
The curious case of the equilateral triangle and 3 concentric rings or the hidden angle bisector
Each of the vertices of an equilateral triangle lie on one of the three concentric circles with radii 1, 2 and 3. Find the length of the side of the equilat...
Let’s use the Angle Bisector Theorem extensions!
What is the ratio of the ellipse’s width to its height?#math #maths #mathchat #mathschat #nerdsniping #MTBoS #iteachmath pic.twitter.com/CGntV3U1f3 — Matt En...
Triangle in a Square: Areas
So this happened today:
There’s more to the Angle Bisector Theorem
I just ran into a few very simple extensions of the angle bisector theorem which I’ve never noticed before. Since I couldn’t easily find this anywhere online...
Proof of Pentagon Construction
I was walking through the following construction of a regular pentagon from the AoPS Geometry textbook recently.
Revisiting the 15-75-90
[This is an old post I kept around which seems appropriate to publish today before AMC 8 (which unfortunately makes for poor blogging fodder)]
Golden Ratios in Regular Polygons
Here’s the first 3: for the triangle, square and pentagon.
Visual Proof: The mediant is in the middle
Usually I’ve seen this done algebraically. See: 213-farey-sequences But from a vector addition standpoint (or
6/5 Woven Math
I’ve been wanting to do this math/art project since I first saw Allison’s artwork on twitter. I finally had enough time to practice and find the supplies. O...
3/20 Visible Math
This week started with a walk through of the MathCounts problem that I gave out last week to do at home.
Fun with Pentagons
I’m in the mood for a geometry walk-through. I’ll start out by saying this one has tons of solutions. I’ve thought of 3 or 4 and seen several additional ones...
11/14 2017 AMC 8 and a digression
Math Club was super easy for me today. I paced outside the classroom while everyone took AMC8.
Heron’s Formula
The MathCounts guide for the year arrived today and I was looking over the problems. The following one caught my eye.
Elegant Solutions vs. Hacking
This is a study in contrasts around a fun problem by @eylem:
Angle bisector
I’ve officially reached the point of the Summer where I’m missing interacting with kids besides my own. In the meantime, this is another geometry walk-throu...
Double angles
I’ve been thinking about a generalization of the 15-75-90 construction over the last few days and have realized there are a lot more interesting consequences...
And yet more 15-75-90 fun
Continuing on the theme of 15-75-90 triangles (See: Last time and First Time) several interesting riffs on 15-75-90’s in a box have come up recently.
First Impressions: A Decade of the Berkeley Math Circle
After seeing a recommendation online, this book arrived at the house in the mail yesterday. I started reading it after dinner and was immediately inspired. ...
15 - 75 Triangle Redux
I came up with this problem after looking at the original one from @five_triangles (Find the area of the trapezoid ABCD) That’s a lot of fun but along the way…
5/23 Triangle Conundrum
In the middle of last week, the MOEMS awards for the year arrived. So I started handing out patches and medals. I’m fairly happy with our overall performance...
4/25 Platonic Solids
This week’s inspiration started with a very late school bus. My son’s bus driver has been on vacation and the substitute drivers have been really, really tar...
My own Geometry Puzzle
(This is based on my previous explorations of the @solvemymaths problems. As far as I know its a new so I’m very happy with it. Usually I just collate probl...
Mid-Winter break Geometry
By tradition, I’m going off on some problem solving walk-throughs:
Winter break cyclic 3-4-5
This is another exercise in documenting geometry problem solving. I chose this problem because again it has a 3-4-5 triangle within it and the overall setup ...
1:2 triangles and their link to Pythagorean triples
I’ve mentioned before how instinctively it feels like the 1:2 triangle ought to have a more natural angle measure. In fact its in a 90 - 26.57 - 63.43 degree...
More Geometry in a Box (Trisection)
Continuing an occasional topic, I saw another great simple box construction.
Another Fundamental Square Construction
Its amazing how much for want of a better word beauty is lurking in very simple constructions. I’ve talked about some square variants before: sometimes-one-d...
Summer Odds and Ends
This is mostly a process update during the quiet days of summer.
Memorial Day Miscellania
Pedagogy Riff
2 Fun 3-4-5 Pythagorean Triples
I’m going to warehouse these problems from @five_triangles here. I really like how they both show constructions for a 3-4-5 Pythagorean Triple. My plan is t...
Geometry - Similar Triangles again
For spring break here’s a geometry walk through I wrote up a while back but never got around to publishing.
Looking for a more elegant approach
This is a continuation of my geometry problem solving posts. I spent the last few days thinking about the above problem from @five_triangles. This ends up…
2/23 Prepping for Pi Day
Riff on the what I want the group to focus on I was reading a post on the natural math site: http://naturalmath.com/math-circles-1001-leaders-course/ around...
President’s Day Break Geometry
There’s no school on Tuesday for President’s day and therefore no Math Club. If I had realized this more fully I would have perhaps picked an extra problem o...
Sometimes its Harder the other way Part 2
I spent some time thinking about what initially looked like a very simple triangle congruence problem last night which I’ve outlined below. Given a perpendic...
Geometry Diversion / corny geometry pun.
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Using Symmetry: More complex proof walk through
This is a continuation of the last 2 posts exploring using symmetry in proofs. This example is the most complicated yet. Symmetry via reflection produced the...
Using Symmetry: Geometry Walk Though #2
This is another in my series exploring geometric problem solving through rigid transformations (and creating diagrams with geogebra). See: using-symmetry-rat...
Using Symmetry: Ratios in square problem
I’ve been trying to put together a tutorial for working on geometry problems over the summer based on the various problems I’ve tried out. The outline is onl...
Sometimes one direction is a lot tricker than the other
I was looking at a twitter post yesterday:
The Geometric Mean
I’ve been thinking a bit about the geometric mean this week after it turned up in two separate
Random Geometry Recursion
You can recursively keep halving the length of the diagonals in this fashion. Creating a series of smaller 30-60-90 triangles in the process. (I promise I ...
Geometric Translations
Recently inspired by some posts on problem solving over at problemproblems.wordpress.com I’ve been combing through the geometry problems at gogeometry.blogsp...
6/2 Pentagrams and some inspirational math.
Just in the nick of time for the end of the year, I received the notice that Jo Boaler’s Week of Inspirational Math. https://www.youcubed.org/week-of-inspira...
The problem I’m thinking about right now.
I worked with my son on this problem last night from AMC8 and I think its really interesting and plays off the previous pentagon problem I had given to the k...
Cool Geometry Problem
I ran into another problem that I think demonstrates an unexpected and interesting fact
Virtual Math Club
Tomorrow our district is shutting down while the teachers go on a one day walk out. As a result there will be no math club session. But that’s no problem - ...
Simplifying a Proof
I found this problem online yesterday (briefly) via a post on the google+ k-12 education. In the figure below
3/10 Pi Day (approximately)
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