Tuesday, January 3, 2017

Inscribed Circle (Euclidea series)

Sometime in the past two years, Sue VanHattum, introduced us to Euclid the Game. This is a nice series of classical construction puzzles (compass and straight-edge) built on top of Geogebra. We recently returned to it and saw a link to another version that we've been playing a lot recently:

In EtG, there is a nice discussion in the comments section. Euclidea doesn't have this feature, so I decided to write blog posts chronicling some of our struggles and, hopefully, starting a place for discussion. I'm not planning to write about every challenge or post answers to all of them, but am happy to take requests. Otherwise, I'll write about the puzzles I find challenging and/or interesting for some reason.

Alpha Pack

I think there are two things worth talking about in alpha pack: (1) the puzzle that has stumped us and (2) the location of the V stars.

That darned square (or is it a diamond?)
The last puzzle in the alpha pack is the one that has stumped us. Our target is this inscribed square:

The kicker is to construct it in 7 elementary moves!

Our thought process
We need four line moves to draw the sides of the square. That means we have only 3 moves to find the other three vertices. We can get one by drawing the diameter of the circle, so we have two moves to find the other two vertices.

We can easily find those two side vertices with three elementary moves, but are really stuck on the idea needed to get one less move.

Some spoilers

Six move square
Our approach to get the construction in 8 elementary moves serves easily to get the 6 move construction. Below, I've included the finished picture which should be enough to see the approach (it isn't very involved anyway).

We found V stars in the following puzzles: equilateral triangle tutorial, 60 degree angle (1.1), and rhombus in a rectangle (1.5).

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