# Beautiful planar geometry theorems not encountered in high school

I would like to impress college students (undergraduates in the U.S.) that there is more to planar geometry beyond what they learned in high school. I would like to show them beautiful theorems they likely never encountered. They surely know that triangle angle bisectors meet in a point, but quite likely they do not know Morley's trisection theorem: (Image from Zach Abel.)
But this is, in some sense, my sole example. So:

Q. What are some other beautiful and perhaps surprising theorems in plane geometry that students would not likely have encountered in a high-school geometry curriculum?

I am grateful for the many wonderful answers:

• Theorems of Ceva and Menelaus and Desargues
• Theorems of Ptolemy and Pappus
• Steiner's porism
• Van Aubel's theorem
• Circle power theorems
• Inversions in a circle
• Poncelet's porism
• The Euler line
• Pappus's Hexagon theorem
• Napoleon's theorem
• Chapple: altitudes concur
• The circle power theorems were taught in my h.s. geometry class but they are not taught in all h.s. geometry classes. Not sure how widely they are or are not taught though. One of the high points of 1978 for me... – user1027 Nov 16 '19 at 12:18
• I never did anything with inversions in a circle in high school, but I find all of that stuff to be really beautiful. I think it would be accessible in high school, and the connection to stereographic projection of the sphere gives a nice opportunity to break into the third dimension. – Steven Gubkin Nov 16 '19 at 16:03
• By all means introduce these theorems with dynamic geometry software - Cinderella (my favorite) or any of the others - many of which easily run on tablets or smartphones. Cinderella, e.g., does inversions which means Steiner's porism and so much more can be so beautifully created and manipulated (e.g., create in non inverted space identical circles tangent to each other and parallel lines, then see the inversion!). Seeing the relationships hold as the points/lines/angles/whatever are dragged around makes all of these theorems totally real. – davidbak Nov 16 '19 at 23:51
• @davidbak: I've been using Cinderella for years, but now I feel I have to learn Geogebra. I would be interested in a comparison. – Joseph O'Rourke Nov 17 '19 at 1:07

Given that I had a good time coercing Mathematica to give me the pictures below I might as well promote my comment to an answer.

My favorite such a result is the so called Steiner's porism.

It is a statement about the relation of two circles, one inside the other. Such as these two: We get a so called Steiner chain of circles from this by first drawing a circle that is tangent to both the given circles. We have a lot of freedom in choosing exactly where we draw this new circle (=the first circle of the chain). Then we add a fourth circle (=the second of the cain) that is tangent to the two circles we started with as well as the freshly added third circle. Next we draw a circle tangent to both the two circles we started with as well as the fourth circle, then we draw a circle tangent to circles number 1,2 and 5 etc. We end up with a picture like the one below, where a chain of seven circles, all red, all tangent to the two we started with. You observe that this chain has a very special property. The seventh circle of the chain happens to be tangent to the first circle of the chain. This says something very special about the relation of the two black circles we started with. Normally it does not happen, but I designed the picture very carefully to make it so.

Initially you might expect that whether this miracle happens depends on where we drew that first circle of the chain. The somewhat surprising result is that this is not the case. Behold! Here's another chain of seven circles. This time all blue. And it also closes after the seventh circle in the chain has been added. Steiner's porism states that this is always the case. The Steiner chain either always closes (with the last circle tangent to the first) after a fixed number of steps (depending on the relation of the two black circles), possibly only after a number of laps, or it never does. Irrespective of the choice of the position of the first circle of the chain. So for our two black circles the chain always closes after the seventh.

The proof is surprisingly simple. That is, after you have covered the basic properties of circle inversions (a theme suggested also by Steven Gubkin in the comments). Namely, you surely agree that the result would be trivial, if the two black circles were concentric. The point is that we can always find an inversion that makes that happen. Furthermore, inversions map circles to circles (or lines as a degenerate case), and preserve tangency.

• Even though I studied a bit more plane geometry on my own when in high school. I only learned about inversions in IMO training. IIRC it never came when I was competing, but that was not necessarily the point :-) – Jyrki Lahtonen Nov 16 '19 at 21:43
• Beautiful! ${}$ – Joseph O'Rourke Nov 17 '19 at 1:09
• Sorry about the bump. I realized that an animation is better than a static of the last frame. Also, some of the circles had jaggies. They were created as polygons connecting 120 points. Applying an inversion made them unevenly distributed, hence the jagginess. I simply doubled the point count to 240. Might have been better to recalculate the whole circle, but ... I was simply testing the functions. – Jyrki Lahtonen Nov 18 '19 at 8:39

Although I was shown the theorems of Ceva and Menelaus in high school, I don't think they're part of the standard curriculum. And I consider them beautiful, especially when considered together so that their similarities become apparent.

I'd also be inclined to nominate Desargues's theorem, especially in view of the observation that the Desargues configuration exhibits the theorem 10 times over: You can start with any of the configuration's 10 points, consider the configuration's 3 lines through that point, find two triangles in the configuration with their vertices on those 3 lines, find (still in the configuration) the intersections of corresponding sides of the triangles, and see that those three intersection points lie on one of the configuration's lines.

• Definitely Ceva & Menelaus & Desargues all beyond high-school, and each with its own beauty. Here is an attempt to unify Ceva & Menelaus, by: Branko Ćurgus. – Joseph O'Rourke Nov 16 '19 at 2:14
• Dumbing down takes its toll. All of these (and Ptolemy and Pappus, but not the Morley theorem) were taught in the UK when I was in secondary school - which was quite a long time ago! – alephzero Nov 16 '19 at 14:09
• I personally like Desargues theorem especially when you start with two triangles not in the same plane and build a little model with straws (or a big model with 6' dowels or pieces of bamboo) - the theorem is so simple and so pretty once you see it that way! And then you can move into projective geometry with all the wonderful dualisms (and simpler proofs due to no parallel lines). – davidbak Nov 16 '19 at 23:43

Not quite as beautiful as Morley’s trisection theorem, but here are two I never saw in high school but which I find beautiful:

• Ptolemy’s theorem, which says that for a cyclic quadrilateral with vertices on a circle, the sum of the products of opposite sides is equal to the product of the diagonals. • A theorem of Pappus, which shows how to construct the arithmetic mean, geometric mean and harmonic mean of two lengths in one semi-circular diagram. • I wrote my master's thesis on how intuitive oblique-angle trigonometry could be if we brought Ptolemy's theorem down to the high school level. – Matthew Daly Nov 16 '19 at 10:12
• @MatthewDaly Can you link to your thesis? – Steven Gubkin Nov 16 '19 at 16:00
• The syntax for getting the alt text to actually show up as the mouseover text is somewhat annoying and not well documented for some reason in the "?" that gives markup help, so I made that happen in this case. – kcrisman Nov 17 '19 at 2:33
• Ptolemy's inequality also has a beautiful proof with circle inversion. – Steven Gubkin Nov 18 '19 at 15:23

For the "beautiful" aspect, you might be inspired by other geometry theorems as illustrated by the 1960's painter Crockett Johnson. For example, here is the Morley Triangle, which is the first place I ever saw that particular theorem. Here is his painting of Pascal's Hexagon: When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points. Cross-Ratio in a Conic: "French mathematician Michel Chasles introduced a result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio.This opened up an entirely different way of describing conic sections." Pappus's Hexagon Theorem has a particularly nice diagram associated with it. Here, $$A$$, $$B$$ and $$C$$ can be any three points along the line $$g$$, while $$a$$, $$b$$ and $$c$$ can be any three points along the line $$h$$. The theorem states that the points $$X$$, $$Y$$ and $$Z$$ at the intersections all lie along a common line.

This theorem was known in antiquity. What is particularly cool is that we can apply the duality principle of projective geometry (which allows us to exchange the roles of points and lines) to get a completely new theorem. The basic principles of duality are

• line $$\leftrightarrow$$ point
• intersection of two lines $$\leftrightarrow$$ line through two points
• three lines meeting at a point $$\leftrightarrow$$ three points along the same line

In the dual version of the theorem, the lines $$GA$$, $$GB$$ and $$GC$$ take on the roles of the points $$A$$, $$B$$ and $$C$$ in the original, while the lines $$Ha$$, $$Hb$$ and $$Hc$$ take on the roles of the points $$a$$, $$b$$ and $$c$$. The points $$G$$ and $$H$$ take on the roles of the lines $$g$$ and $$h$$ from the original: just as $$A$$, $$B$$ and $$C$$ all lay on the line $$g$$ in the original, so do the lines $$GA$$, $$GB$$ and $$GC$$ meet at the point $$G$$.

If we continue to apply these principles to the first diagram, then we arrive at the second diagram. The dual theorem is that the three coloured lines $$X$$, $$Y$$ and $$Z$$ all meet at a single point $$U$$, just as the three points $$X$$, $$Y$$ and $$Z$$ all lay on a common line $$u$$ in the original diagram.

I believe (though I am not sure) that this second version was discovered in exactly this way, by applying duality to the first version over a thousand years after the first version was discovered.

• Note that this is a special case of Pascal's theorem, also beautiful. – user13050 Nov 18 '19 at 22:53

### 1. Fundamental theorem of transformational plane geometry

A transformation $$\psi$$ is an isometry iff $$\psi$$ can be expressed as a composition of three or fewer reflections.

### 2. Laisant's theorem Given a triangle $$ABC$$, extend the two sides $$AB$$ and $$AC$$ as necessary to point $$A_c$$ on $$AB$$ at distance $$|BC|$$ from $$B$$ and $$C_b$$ on $$AC$$ at distance $$|BC|$$ from $$C$$ such that both points are on the same side of $$BC$$ as $$A$$. Then the direction of the line $$A_bA_c$$ is independent of $$BC$$.

### 3. Dandelin spheres

An ellipse is a plane section of a cone. It is possible to fit one sphere into the cone to touch the plane, between the plane and the vertex, and another sphere to touch the plane and the cone on the other side.

Dandelin, a professor of mechanics at Liege University, proved that the two spheres touch the ellipse at its foci, and that the directrices of the ellipse are the lines in which the cutting plane meets the planes of the circles in which the spheres touch the cone. ### 4. Eyeball theorem

The tangents to each of two circles from the centre of the other are drawn. Then the line segments $$AB$$ and $$XY$$ are equal in length. ### 5. Platonic solids

There are only five convex regular polyhedra, known as the Platonic solids.

### 6. Dijkstra's theorem

In any triangle $$ABC$$ we have $$\mathrm{sgn}(\alpha+\beta-\gamma)=\mathrm{sgn}(a^2+b^2-c^2)$$.

Here $$\mathrm{sgn}$$ is the signum function. It is easy to deduce from the law of cosines but Dijkstra proved it more simply. The pythagorean theorem is a special case.

I'd recommend literally anything in either of these books, but I'm also partial to Napoleon's theorem. In particular, we can prove:

• The equilateral triangles erected outward on the sides $$a,\,b,\,c$$ of an area-$$\Delta$$ triangle have orthocetres at the vertices of an equilateral triangle of squared side length $$\frac{a^2+b^2+c^2}{6}+\frac{2\Delta}{\sqrt{3}}$$;
• If we erect inward instead, the resulting equilateral triangle has squared side length $$\frac{a^2+b^2+c^2}{6}-\frac{2\Delta}{\sqrt{3}}$$;
• The equilateral triangles with squared side lengths $$\frac{a^2+b^2+c^2}{6}\pm\frac{2\Delta}{\sqrt{3}}$$ differ in area by $$\Delta$$.

The smaller such triangle having area $$\ge0$$ is Weitzenböck's inequality.

Van Aubel's theorem comes to mind for me:

On each side of a planar quadrilateral (can be non-convex), construct a square (all external or all internal to the quadrilateral). If we construct line segments between the centers of opposite squares, then the two line segments are orthogonal to each other and have equal length.

I'm posting this for Joseph Malkevitch, who had difficulty while posting. He wanted to include one of his favorite theorems, the Wallace-Bolyai-Gerwien theorem, that says that any two polygons of the same area have a common dissection. See this impressive app developed by Satyan Devadoss's students:

Scissors Congruence app Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth. The radius of the limiting circle is called the Kepler-Beoucamp constant. • What's the theorem? Surely it's not just that this is scale-independent? – Peter Taylor Dec 11 '19 at 8:18
• @PeterTaylor, maybe that the constant is positive? – LSpice Jun 19 at 20:30

A triangle's altitudes concur.

It seems that this theorem wasn't known until Chapple.

Related to this:

If $$H$$ is $$ABC$$'s orthocentre, $$A, B, C, H$$ are an orthocentric system, i.e. the straight line between any two of them is perpendicular to that between the other two; and each is the orthocentre of the triangle formed by the other three.

Where $$R$$ is $$ABC$$'s circumradius, $$AH=2R\cos\angle BAC$$. (And similarly for $$BH$$ and $$CH$$.)

The hyperboloid of one sheet appears to be very curvaceous everywhere, and yet it is actually a ruled surface (i.e., can be traced out from a straight line moving in space).