# Revisiting arc midpoints in complex numbers

## 1. Synopsis

One of the major headaches of using complex numbers in olympiad geometry problems is dealing with square roots. In particular, it is nontrivial to express the incenter of a triangle inscribed in the unit circle in terms of its vertices.

The following lemma is the standard way to set up the arc midpoints of a triangle. It appears for example as part (a) of Lemma 6.23.

Theorem 1 (Arc midpoint setup for a triangle)

Let ${ABC}$ be a triangle with circumcircle ${\Gamma}$ and let ${M_A}$, ${M_B}$, ${M_C}$ denote the arc midpoints of ${\widehat{BC}}$ opposite ${A}$, ${\widehat{CA}}$ opposite ${B}$, ${\widehat{AB}}$ opposite ${C}$.

Suppose we view ${\Gamma}$ as the unit circle in the complex plane. Then there exist complex numbers ${x}$, ${y}$, ${z}$ such that ${A = x^2}$, ${B = y^2}$, ${C = z^2}$, and

$\displaystyle M_A = -yz, \quad M_B = -zx, \quad M_C = -xy.$

Theorem 1 is often used in combination with the following lemma, which lets one assign the incenter the coordinates ${-(xy+yz+zx)}$ in the above notation.

Lemma 2 (The incenter is the orthocenter of opposite arc midpoints)

Let ${ABC}$ be a triangle with circumcircle ${\Gamma}$ and let ${M_A}$, ${M_B}$, ${M_C}$ denote the arc midpoints of ${\widehat{BC}}$ opposite ${A}$, ${\widehat{CA}}$ opposite ${B}$, ${\widehat{AB}}$ opposite ${C}$. Then the incenter of ${\triangle ABC}$ coincides with the orthocenter of ${\triangle M_A M_B M_C}$.

Unfortunately, the proof of Theorem 1 in my textbook is wrong, and I cannot find a proof online (though I hear that Lemmas in Olympiad Geometry has a proof). So in this post I will give a correct proof of Theorem 1, which will hopefully also explain the mysterious introduction of the minus signs in the theorem statement. In addition I will give a version of the theorem valid for quadrilaterals.

## 2. A Word of Warning

I should at once warn the reader that Theorem 1 is an existence result, and thus must be applied carefully.

To see why this matters, consider the following problem, which appeared as problem 1 of the 2016 JMO.

Example 3 (JMO 2016, by Zuming feng)

The isosceles triangle ${\triangle ABC}$, with ${AB=AC}$, is inscribed in the circle ${\omega}$. Let ${P}$ be a variable point on the arc ${BC}$ that does not contain ${A}$, and let ${I_B}$ and ${I_C}$ denote the incenters of triangles ${\triangle ABP}$ and ${\triangle ACP}$, respectively. Prove that as ${P}$ varies, the circumcircle of triangle ${\triangle PI_{B}I_{C}}$ passes through a fixed point.

By experimenting with the diagram, it is not hard to guess that the correct fixed point is the midpoint of arc ${\widehat{BC}}$, as seen in the figure below. One might be tempted to write ${A = x^2}$, ${B = y^2}$, ${C = z^2}$, ${P = t^2}$ and assert the two incenters are ${-(xy+yt+xt)}$ and ${-(xz+zt+xt)}$, and that the fixed point is ${-yz}$.

This is a mistake! If one applies Theorem 1 twice, then the choices of “square roots” of the common vertices ${A}$ and ${P}$ may not be compatible. In fact, they cannot be compatible, because the arc midpoint of ${\widehat{AP}}$ opposite ${B}$ is different from the arc midpoint of ${\widehat{AP}}$ opposite ${C}$.

In fact, I claim this is not a minor issue that one can work around. This is because the claim that the circumcircle of ${\triangle P I_B I_C}$ passes through the midpoint of arc ${\widehat{BC}}$ is false if ${P}$ lies on the arc on the same side as ${A}$! In that case it actually passes through ${A}$ instead. Thus the truth of the problem really depends on the fact that the quadrilateral ${ABPC}$ is convex, and any attempt with complex numbers must take this into account to have a chance of working.

## 3. Proof of the theorem for triangles

Fix ${ABC}$ now, so we require ${A = x^2}$, ${B = y^2}$, ${C = z^2}$. There are ${2^3 = 8}$ choices of square roots ${x}$, ${y}$, ${z}$ we can take (differing by a sign); we wish to show one of them works.

We pick an arbitrary choice for ${x}$ first. Then, of the two choices of ${y}$, we pick the one such that ${-xy = M_C}$. Similarly, for the two choices of ${z}$, we pick the one such that ${-xz = M_B}$. Our goal is to show that under these conditions, we have ${M_A = -yz}$ again.

The main trick is to now consider the arc midpoint ${\widehat{BAC}}$, which we denote by ${L}$. It is easy to see that:

Lemma 4 (The isosceles trapezoid trick)

We have ${\overline{AL} \parallel \overline{M_B M_C}}$ (both are perpendicular to the ${\angle A}$ bisector). Thus ${A L M_B M_C}$ is an isosceles trapezoid, and so ${ A \cdot L = M_B \cdot M_C }$.

Thus, we have

$\displaystyle L = \frac{M_B M_C}{A} = \frac{(-xz)(-xy)}{x^2} = +yz.$

Thus

$\displaystyle M_A = -L = -yz$

as desired.

From this we can see why the minus signs are necessary.

Exercise 5

Show that Theorem 1 becomes false if we try to use ${+yz}$, ${+zx}$, ${+xy}$ instead of ${-yz}$, ${-zx}$, ${-xy}$.

## 4. A version for quadrilaterals

We now return to the setting of a convex quadrilateral ${ABPC}$ that we encountered in Example 3. Suppose we preserve the variables ${x}$, ${y}$, ${z}$ that we were given from Theorem 1, but now add a fourth complex number ${t}$ with ${P = t^2}$. How are the new arc midpoints determined? The following theorem answers this question.

Theorem 6 (${xytz}$ setup)

Let ${ABPC}$ be a convex quadrilateral inscribed in the unit circle of the complex plane. Then we can choose complex numbers ${x}$, ${y}$, ${z}$, ${t}$ such that ${A = x^2}$, ${B = y^2}$, ${C = z^2}$, ${P = t^2}$ and:

• The opposite arc midpoints ${M_A}$, ${M_B}$, ${M_C}$ of triangle ${ABC}$ are given by ${-yz}$, ${-zx}$, ${-xy}$, as before.
• The midpoint of arc ${\widehat{BP}}$ not including ${A}$ or ${C}$ is given by ${+yt}$.
• The midpoint of arc ${\widehat{CP}}$ not including ${A}$ or ${B}$ is given by ${-zt}$.
• The midpoint of arc ${\widehat{ABP}}$ is ${-xt}$ and the midpoint of arc ${\widehat{ACP}}$ is ${+xt}$.

This setup is summarized in the following figure.

Note that unlike Theorem 1, the four arcs cut out by the sides of ${ABCP}$ do not all have the same sign (I chose ${\widehat{BP}}$ to have coordinates ${+yt}$). This asymmetry is inevitable (see if you can understand why from the proof below).

Proof: We select ${x}$, ${y}$, ${z}$ with Theorem 1. Now, pick a choice of ${t}$ such that ${+yt}$ is the arc midpoint of ${\widehat{BP}}$ not containing ${A}$ and ${C}$. Then the arc midpoint of ${\widehat{CP}}$ not containing ${A}$ or ${B}$ is given by

$\displaystyle \frac{z^2}{-yz} \cdot (+yt) = -zt.$

On the other hand, the calculation of ${-xt}$ for the midpoint of ${\widehat{ABP}}$ follows by applying Lemma 4 again. (applied to triangle ${ABP}$). The midpoint of ${\widehat{ACP}}$ is computed similarly. $\Box$

In other problems, the four vertices of the quadrilateral may play more symmetric roles and in that case it may be desirable to pick a setup in which the four vertices are labeled ${ABCD}$ in order. By relabeling the letters in Theorem 6 one can prove the following alternate formulation.

Corollary 7

Let ${ABCD}$ be a convex quadrilateral inscribed in the unit circle of the complex plane. Then we can choose complex numbers ${a}$, ${b}$, ${c}$, ${d}$ such that ${A = a^2}$, ${B = b^2}$, ${C = c^2}$, ${D = d^2}$ and:

• The midpoints of ${\widehat{AB}}$, ${\widehat{BC}}$, ${\widehat{CD}}$, ${\widehat{DA}}$ cut out by the sides of ${ABCD}$ are ${-ab}$, ${-bc}$, ${-cd}$, ${+da}$.
• The midpoints of ${\widehat{ABC}}$ and ${\widehat{BCD}}$ are ${+ac}$ and ${+bd}$.
• The midpoints of ${\widehat{CDA}}$ and ${\widehat{DAB}}$ are ${-ac}$ and ${-bd}$.

To test the newfound theorem, here is a cute easy application.

Example 8 (Japanese theorem for cyclic quadrilaterals)

In a cyclic quadrilateral ${ABCD}$, the incenters of ${\triangle ABC}$, ${\triangle BCD}$, ${\triangle CDA}$, ${\triangle DAB}$ are the vertices of a rectangle.

## 4 thoughts on “Revisiting arc midpoints in complex numbers”

1. Here’s my preferred way to think about this: let $A_1A_2\dots A_n$ be a cyclic $n$-gon on the unit circle with $n\geq3$, and let $B_{1,2},B_{2,3},\dots,B_{n,1}$ be the midpoints of the $n$ arcs partitioning the circle.

Key Observation: if $A_1,A_2,\dots,A_n$ go counterclockwise around the circle, then the product $\prod_i B_{i,i+1} A_i^{-1}$ is $\exp(i\pi) = -1$, because the sum of the corresponding arcs is precisely half the circumference.

So, if $a_1$ is an arbitrary square root of $A_1$, then there is a **unique** square root $a_2$ of $A_2$ such that $B_{1,2} = -a_1a_2$: namely, $a_2 = -B_{1,2} a_1^{-1}$ works! Continuing in this manner uniquely specifies all $a_i$. The catch is, the choice is consistent if and only if $n$ is odd: we require $(-1)^n$ to be consistent with $-1$ from the Key Observation.

The case $n$ is even (e.g. quadrilateral as you discuss) is similar, except we can no longer take all the signs in the $B_{i,i+1} = \pm a_ia_{i+1}$ to be $-1$. An odd number of the signs must be $+1$ instead.

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• In the explanation for the cyclic inductive construction, I should have elaborated that the terms $B_{i,i+1} A_i^{-1} = \pm a_{i+1} a_i^{-1}$ form a cyclic telescoping product (I only mentioned the sign issues). In any case, an example where I needed to think through this carefully was http://artofproblemsolving.com/community/c6h404946p2271784

(Sorry I forgot about WordPress math in the previous comment, and I can’t seem to edit.)

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2. No comments on the post, but a question: what do you use for writing WordPress math? I am struggling with open-source solutions. Thank you in advance!

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