Subfield: Geometry


Question: Let $A B C$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $B C, C A, A B$ intersect $\ell$ at the points $X, Y, Z$ different from $P$. Prove that the circumcircles of the triangles $A X P, B Y P$ and $C Z P$ have a common point different from $P$ or are mutually tangent at $P$.

Solution:
Let $\omega_{A}, \omega_{B}, \omega_{C}$ and $\omega$ be the circumcircles of triangles $A X P, B Y P, C Z P$ and $A B C$ respectively. The strategy of the proof is to construct a point $Q$ with the same power with respect to the four circles. Then each of $P$ and $Q$ has the same power with respect to $\omega_{A}, \omega_{B}, \omega_{C}$ and hence the three circles are coaxial. In other words they have another common point $P^{\prime}$ or the three of them are tangent at $P$.

We first give a description of the point $Q$. Let $A^{\prime} \neq A$ be the second intersection of $\omega$ and $\omega_{A}$; define $B^{\prime}$ and $C^{\prime}$ analogously. We claim that $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ have a common point. Once this claim is established, the point just constructed will be on the radical axes of the three pairs of circles $\left\{\omega, \omega_{A}\right\},\left\{\omega, \omega_{B}\right\},\left\{\omega, \omega_{C}\right\}$. Hence it will have the same power with respect to $\omega, \omega_{A}, \omega_{B}, \omega_{C}$.

<img_3996>

We proceed to prove that $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ intersect at one point. Let $r$ be the circumradius of triangle $A B C$. Define the points $X^{\prime}, Y^{\prime}, Z^{\prime}$ as the intersections of $A A^{\prime}, B B^{\prime}, C C^{\prime}$ with $\ell$. Observe that $X^{\prime}, Y^{\prime}, Z^{\prime}$ do exist. If $A A^{\prime}$ is parallel to $\ell$ then $\omega_{A}$ is tangent to $\ell$; hence $X=P$ which is a contradiction. Similarly, $B B^{\prime}$ and $C C^{\prime}$ are not parallel to $\ell$.

From the powers of the point $X^{\prime}$ with respect to the circles $\omega_{A}$ and $\omega$ we get

$$
X^{\prime} P \cdot\left(X^{\prime} P+P X\right)=X^{\prime} P \cdot X^{\prime} X=X^{\prime} A^{\prime} \cdot X^{\prime} A=X^{\prime} O^{2}-r^{2}
$$

hence

$$
X^{\prime} P \cdot P X=X^{\prime} O^{2}-r^{2}-X^{\prime} P^{2}=O P^{2}-r^{2}
$$

We argue analogously for the points $Y^{\prime}$ and $Z^{\prime}$, obtaining

$$
X^{\prime} P \cdot P X=Y^{\prime} P \cdot P Y=Z^{\prime} P \cdot P Z=O P^{2}-r^{2}=k^{2} .
\tag{1}
$$

In these computations all segments are regarded as directed segments. We keep the same convention for the sequel.

We prove that the lines $A A^{\prime}, B B^{\prime}, C C^{\prime}$ intersect at one point by CEvA's theorem. To avoid distracting remarks we interpret everything projectively, i. e. whenever two lines are parallel they meet at a point on the line at infinity.



Let $U, V, W$ be the intersections of $A A^{\prime}, B B^{\prime}, C C^{\prime}$ with $B C, C A, A B$ respectively. The idea is that although it is difficult to calculate the ratio $\frac{B U}{C U}$, it is easier to deal with the cross-ratio $\frac{B U}{C U} / \frac{B X}{C X}$ because we can send it to the line $\ell$. With this in mind we apply MENELAUs' theorem to the triangle $A B C$ and obtain $\frac{B X}{C X} \cdot \frac{C Y}{A Y} \cdot \frac{A Z}{B Z}=1$. Hence Ceva's ratio can be expressed as

$$
\frac{B U}{C U} \cdot \frac{C V}{A V} \cdot \frac{A W}{B W}=\frac{B U}{C U} / \frac{B X}{C X} \cdot \frac{C V}{A V} / \frac{C Y}{A Y} \cdot \frac{A W}{B W} / \frac{A Z}{B Z}
$$

<img_3503>

Project the line $B C$ to $\ell$ from $A$. The cross-ratio between $B C$ and $U X$ equals the cross-ratio between $Z Y$ and $X^{\prime} X$. Repeating the same argument with the lines $C A$ and $A B$ gives

$$
\frac{B U}{C U} \cdot \frac{C V}{A V} \cdot \frac{A W}{B W}=\frac{Z X^{\prime}}{Y X^{\prime}} / \frac{Z X}{Y X} \cdot \frac{X Y^{\prime}}{Z Y^{\prime}} / \frac{X Y}{Z Y} \cdot \frac{Y Z^{\prime}}{X Z^{\prime}} / \frac{Y Z}{X Z}
$$

and hence

$$
\frac{B U}{C U} \cdot \frac{C V}{A V} \cdot \frac{A W}{B W}=(-1) \cdot \frac{Z X^{\prime}}{Y X^{\prime}} \cdot \frac{X Y^{\prime}}{Z Y^{\prime}} \cdot \frac{Y Z^{\prime}}{X Z^{\prime}}
$$

The equations (1) reduce the problem to a straightforward computation on the line $\ell$. For instance, the transformation $t \mapsto-k^{2} / t$ preserves cross-ratio and interchanges the points $X, Y, Z$ with the points $X^{\prime}, Y^{\prime}, Z^{\prime}$. Then

$$
\frac{B U}{C U} \cdot \frac{C V}{A V} \cdot \frac{A W}{B W}=(-1) \cdot \frac{Z X^{\prime}}{Y X^{\prime}} / \frac{Z Z^{\prime}}{Y Z^{\prime}} \cdot \frac{X Y^{\prime}}{Z Y^{\prime}} / \frac{X Z^{\prime}}{Z Z^{\prime}}=-1
$$

We proved that CEva's ratio equals -1 , so $A A^{\prime}, B B^{\prime}, C C^{\prime}$ intersect at one point $Q$.
First we prove that there is an inversion in space that takes $\ell$ and $\omega$ to parallel circles on a sphere. Let $Q R$ be the diameter of $\omega$ whose extension beyond $Q$ passes through $P$. Let $\Pi$ be the plane carrying our objects. In space, choose a point $O$ such that the line $Q O$ is perpendicular to $\Pi$ and $\angle P O R=90^{\circ}$, and apply an inversion with pole $O$ (the radius of the inversion does not matter). For any object $\mathcal{T}$ denote by $\mathcal{T}^{\prime}$ the image of $\mathcal{T}$ under this inversion.

The inversion takes the plane $\Pi$ to a sphere $\Pi^{\prime}$. The lines in $\Pi$ are taken to circles through $O$, and the circles in $\Pi$ also are taken to circles on $\Pi^{\prime}$.

<img_3897>

Since the line $\ell$ and the circle $\omega$ are perpendicular to the plane $O P Q$, the circles $\ell^{\prime}$ and $\omega^{\prime}$ also are perpendicular to this plane. Hence, the planes of the circles $\ell^{\prime}$ and $\omega^{\prime}$ are parallel.

Now consider the circles $A^{\prime} X^{\prime} P^{\prime}, B^{\prime} Y^{\prime} P^{\prime}$ and $C^{\prime} Z^{\prime} P^{\prime}$. We want to prove that either they have a common point (on $\Pi^{\prime}$ ), different from $P^{\prime}$, or they are tangent to each other.

<img_3745>

The point $X^{\prime}$ is the second intersection of the circles $B^{\prime} C^{\prime} O$ and $\ell^{\prime}$, other than $O$. Hence, the lines $O X^{\prime}$ and $B^{\prime} C^{\prime}$ are coplanar. Moreover, they lie in the parallel planes of $\ell^{\prime}$ and $\omega^{\prime}$. Therefore, $O X^{\prime}$ and $B^{\prime} C^{\prime}$ are parallel. Analogously, $O Y^{\prime}$ and $O Z^{\prime}$ are parallel to $A^{\prime} C^{\prime}$ and $A^{\prime} B^{\prime}$.

Let $A_{1}$ be the second intersection of the circles $A^{\prime} X^{\prime} P^{\prime}$ and $\omega^{\prime}$, other than $A^{\prime}$. The segments $A^{\prime} A_{1}$ and $P^{\prime} X^{\prime}$ are coplanar, and therefore parallel. Now we know that $B^{\prime} C^{\prime}$ and $A^{\prime} A_{1}$ are parallel to $O X^{\prime}$ and $X^{\prime} P^{\prime}$ respectively, but these two segments are perpendicular because $O P^{\prime}$ is a diameter in $\ell^{\prime}$. We found that $A^{\prime} A_{1}$ and $B^{\prime} C^{\prime}$ are perpendicular, hence $A^{\prime} A_{1}$ is the altitude in the triangle $A^{\prime} B^{\prime} C^{\prime}$, starting from $A$.

Analogously, let $B_{1}$ and $C_{1}$ be the second intersections of $\omega^{\prime}$ with the circles $B^{\prime} P^{\prime} Y^{\prime}$ and $C^{\prime} P^{\prime} Z^{\prime}$, other than $B^{\prime}$ and $C^{\prime}$ respectively. Then $B^{\prime} B_{1}$ and $C^{\prime} C_{1}$ are the other two altitudes in the triangle $A^{\prime} B^{\prime} C^{\prime}$.



Let $H$ be the orthocenter of the triangle $A^{\prime} B^{\prime} C^{\prime}$. Let $W$ be the second intersection of the line $P^{\prime} H$ with the sphere $\Pi^{\prime}$, other than $P^{\prime}$. The point $W$ lies on the sphere $\Pi^{\prime}$, in the plane of the circle $A^{\prime} P^{\prime} X^{\prime}$, so $W$ lies on the circle $A^{\prime} P^{\prime} X^{\prime}$. Similarly, $W$ lies on the circles $B^{\prime} P^{\prime} Y^{\prime}$ and $C^{\prime} P^{\prime} Z^{\prime}$ as well; indeed $W$ is the second common point of the three circles.

If the line $P^{\prime} H$ is tangent to the sphere then $W$ coincides with $P^{\prime}$, and $P^{\prime} H$ is the common tangent of the three circles.