# Are there any list of mathematical constructions which can challenge 12-16 year old students?

Mathematical (geometric) constructions are an interesting way to engage students. It also helps in better understanding of different geometrical properties. For example, Sierpinski triangle or square, Steiner chains. While constructing, it immensely helped students to have a better grasp on geometrical properties.

Are there any list one can refer to? Or propose whatever you think could be tried.

• I love two sites that have gamified construction: euclidthegame.com and sciencevsmagic.net/geo May 21 '21 at 18:13
• What kind of mathematical constructions do you have in mind? It appears that you're only interested in geometric constructions, maybe you can be more specific? May 21 '21 at 20:55

Assuming that by "Geometric constructions", you mean "ruler and compass constructions", here is a sequence of progressively-more-difficult constructions that begin at the trivial and end at the impossible:

1. The "standard" constructions:
1. Perpendicular bisector of a line segment.
2. Angle bisector.
3. Perpendicular to a line through a point.
4. Duplicated line segment.
5. Line parallel to a line through a point.
6. Mirroring a point in a line.
2. Some easy extensions:
1. Line disection (into arbitrarily many pieces)
2. Angles of 30, 45, 60, and 90 degrees.
3. An equilateral triangle.
4. Triangles given:
1. A triangle to copy.
2. Two angles and a side
3. Two sides and the angle between.
4. Three sides
5. The incircle and circumcircle of a triangle.
6. The diameter of a circle.
7. A circle through three points.
8. The tangents to a circle passing through a given point on or outside the circle.
9. A regular square, pentagon, hexagon, octagon, dodecagon (inscribed in a given circle or from one side), or a regular polygon with any (small) number of sides in this list.
10. Any results from Elements.
3. Significantly harder problems:
1. Some of the Wernick List (construct a triangle given three of: the vertices, the circumcenter, the side midpoints, the centroids, the altitude feet, the orthocentre, the feet of the internal angle bisectors, and the incenter - obviously not the unsolvable cases!)
2. The Mohr-Mascheroni constructions.
3. The Poncelet–Steiner constructions.
4. Calculate the best approximation to $$\pi$$ that you can using a ruler and compass.
5. This and this.
4. Trick questions
1. Bisect an angle using only a ruler (not impossible, but the fact that I said "ruler" (as in the physical object) not "straightedge" is important, and it's nothing to do with the markings).
2. Angle trisection, squaring the circle, constructing a regular heptagon.

I would like to add a few of my favourites:

• Apollonian constructions (construct a circle tangent to three objects, where the objects can be points, lines or circles). There are 10 of these problems, some are standard, some are tricky.
• Construct the golden ratio (this is related to the regular pentagon mentioned above).
• Solve a quadratic equation using straightedge and compass (the coefficients and the unit is given as line segments).
• Trisect an angle using paper folding (this cannot be done in general using straightedge and compass).
• Using a piece of string and two pins, one can draw an ellipse. This paper investigates constructions using a tool like this.
• The impossibility of constructing the centre of a circle using only a straightedge.