Geometry is not as popular nowadays as it was. So I was very surprised to learn about a game that features geometry puzzles. It poses problems of Euclidean geometry in the plane, e.g. constructing the tangent to a circle. To solve the problem fully, goals have to be completed. One goal (L) is to do it with the minimum number of steps with all tools available (which include composite tools like the middle perpendicular). Another goal (E) is to do it with the minimum number of elementary tools, which are circles and lines. The third goal (V) is to construct variations of the solution, like reflections of the solution. The goal (E) is often very difficult even for elementary constructions. I show two examples below.

The app is done by some Russian programmers. I could find no information on the authors at all. Actually, there are lots of Russian programmers that make money with their skills in different areas. Besides gaming, an example is flight simulation where Russia programmers made some of the best addons. The programmers are not necessarily skilled geometry geeks. There are old books buried somewhere in libraries that contain lots of geometry problems and solutions. Those are the reminders of the old times when geometry was still popular.

The app is extremely nicely done and shows dedication and skill with user interfaces. Some details could need perfection and I will talk about that later. The app is free. But you can buy more problems. It is a nice touch that you get these for free if you have solved all the previous goals. However, I am considering to donate to these programmers.

Here is an example of a problem. I present the problem with my own program C.a.R. This program could also pose problems with restricted toolsets, called assignments. However, there is no game feature and I never cared about the number of steps taken. It was a program to help to teach geometry in schools.

Your job is to construct the green line given the black lines. In the app, the given objects are not fat and the user cannot set colors. That makes it more difficult than it should be to keep track of the progress.

Of course, it is quite easy to do the job with two given tools, the line tool, and the perpendicular. Moreover, every geometry student should know the following solution.

It takes 5 elementary steps. It is a typical mathematical solution because it is based on another solution to another problem. Mathematics often does that. We know how to construct the middle perpendicular. Thus we know how to construct a perpendicular to a line through a point on the line. It remains to see the tangent as perpendicular to the radius. But the goal was to do it with three elementary steps, not 5. My second solution was the following.

It takes 4 elementary steps still and is based on another well-known geometry fact, the Thales circle. Observe the brown circle (2) to see why this works. It is still not the optimal solution. I now tell you a secret: You can find all solutions on Youtube. Nowadays, going to Youtube is the first step that students take.

The following is the solution with three steps. Unfortunately, I was given that solution before I could find it. Why it works is by no means obvious.

The only proof I could find is a computation with angles.

By basic equivalences, we get that the angles with equal color are equal. By the circle chord theorem, the angle of a chord of a circle with respect to a point on the circle is half the angle with respect to the midpoint of the circle, thus black=2*blue. We also have black+red=90°. Thus black+2*blue=90°.

There should be a simpler proof to this, but I have not yet found it.

All in all, this is an extremely well-done app. I would like to use colors while constructing. I would also like to use the given segments as lines. This is important in the following example, where you have to construct the center of the inscribed circle with only 6 elementary steps. Note that the usual construction takes 4 elementary steps for each angle bisector.

That is already very complicated. The first step is to construct one angle bisector using the blue circles. The second bisector is found with the mysterious red circle which is constructed by one of the blue circles. Its second intersection with the same blue circle yields the second angle bisector. I found this by trial and error. After all, you can construct the angle bisector in C.a.R. so see where it should be. In the app, you can show the targets too. Did I tall you that you can also find the solution on Youtube?

Does anybody have an easy proof?