Hagge Configurations and a Projective Generalization of Inversion

Abstract

In this article, we provide elementary proofs of two projective generalizations of Hagge’s theorems. We describe Steiner’s correspondence as a projective generalization of inversion.

1. Introduction

The discovery of Hagge’s circle by K. Hagge in 1907 [1] opened new perspectives in classical geometry [2,3,4]. In a recent paper [5], Bradley described two generalizations of Hagge’s theorems. By means of coordinate calculations, first he proved the following:

Theorem 1.

Let a triangle $ABC$ be given in the Euclidean plane. Let D be a point, not lying on the side lines of the triangle, and let Σ be a circle passing through D. If $\Sigma \setminus \left\{D\right\}$ meets the circles $BCD$, $ACD$, and $ABD$ at the points U, V, and W, and meets the lines $\overleftrightarrow{AD}$, $\overleftrightarrow{BD}$, and $\overleftrightarrow{CD}$ at the points X, Y, and Z, respectively, then the lines $\overleftrightarrow{UX}$, $\overleftrightarrow{VY}$, and $\overleftrightarrow{WZ}$ are concurrent. (see Figure 1).

Next, he deduced an essentially projective generalization of the following result:

Theorem 2.

Let a triangle $ABC$ be given in the Euclidean plane. Let D, E, and F be non-colinear points, neither of which lies on a side line of the triangle. If a conic Σ passes through the points D, E, and F, and if $\Sigma \setminus \left\{D\right\}$ meets the conics $BCDEF$, $ACDEF$, and $ABDEF$ at the points U, V, and W, and meets the lines $\overleftrightarrow{AD}$, $\overleftrightarrow{BD}$, and $\overleftrightarrow{CD}$ at the points X, Y, and Z, then the lines $\overleftrightarrow{UX}$, $\overleftrightarrow{VY}$, $\overleftrightarrow{WZ}$ are concurrent. (see Figure 2).

Theorem 2 indeed reduces to Theorem 1 if E and F are the ’circular points at infinity’. In this note, we present synthetic, elementary proofs for both of these theorems. Our proof for Theorem 2 does not rely on Theorem 1 (so we may immediately deduce the first theorem from the second one). The reasoning applied in the proof of Theorem 2 is a substantial refinement of that in the proof of Theorem 1. In fact, we show that Theorem 2 is valid in any Pappian projective plane satisfying Fano’s axiom.

In both proofs, we need the following basic facts from projective geometry.

Let a Pappian plane be given, satisfying Fano’s axiom. Then, we have

(A) The three pairs of opposite sides of a complete quadrangle meet any line (not passing through a vertex) in the three pairs of an involution.

(B) If U, V, W, X, Y, and Z are six points on a conic, then the three lines $\overleftrightarrow{UX}$, $\overleftrightarrow{VY}$, and $\overleftrightarrow{WZ}$ are concurrent, if and only if, (UX), (VY), and (WZ) are pairs of an involution on the conic.

For a proof, we refer to Coxeter’s book [6].

2. An Elementary Proof of Theorem 1

We may interpret the Euclidean plane as a part of its projective closure. The latter is a Pappian plane satisfying Fano’s axiom (in fact, it is isomorphic to the real projective plane).

Apply an inversion of pole D, denoting the images of points and sets by a prime. Then, the sets

$$\left\{D,A^{\prime },X^{\prime }\right\},\left\{D,B^{\prime },Y^{\prime }\right\},\left\{D,C^{\prime },Z^{\prime }\right\},\left\{A^{\prime },B^{\prime },W^{\prime }\right\},\left\{A^{\prime },C^{\prime },V^{\prime }\right\},\left\{B^{\prime },C^{\prime },U^{\prime }\right\}$$

are colinear. Therefore, the opposite sides of the complete quadrangle $A^{\prime }{B}^{\prime }{C}^{\prime }D$ meet the line ${\Sigma }^{\prime }$ at the pairs $\left(U^{\prime }{X}^{\prime }\right)$, $\left(V^{\prime }{Y}^{\prime }\right)$, and $\left(W^{\prime }{Z}^{\prime }\right)$. With (A), these are the three pairs of an involution. On the other hand, inversion preserves cross ratio cr, and pairs $\left(P_1{P}_1^{\prime }\right)$, $\left(P_2{P}_2^{\prime }\right)$, and $\left(P_3{P}_3^{\prime }\right)$ are pairs of an involution, if and only if $\mathrm{cr}(P_1,P_2,P_3,P_3^{\prime })=\mathrm{cr}(P_1^{\prime },P_2^{\prime },P_3^{\prime },P_3)$. Thus, the involution sends the pairs $\left(U^{\prime }{X}^{\prime }\right)$, $\left(V^{\prime }{Y}^{\prime }\right)$, and $\left(W^{\prime }{Z}^{\prime }\right)$ to the pairs of an involution on the circle $\Sigma ={\left({\Sigma }^{\prime }\right)}^{\prime }$. Hence, by (B), the lines

$$\overleftrightarrow{UX},\overleftrightarrow{VY},\overleftrightarrow{WZ}$$

are concurrent. (see Figure 3).

3. An Elementary Proof of Theorem 2

In this section, we consider a Pappian projective plane $\mathbf{P}$ satisfying Fano’s axiom. First, we collect some basic facts concerning the so-called Steiner correspondence. These suggest that it is a good candidate for a purely projective generalization of inversion. Indeed, Steiner correspondence will play the same role in the proof of Theorem 2 as inversion in the proof of Theorem 1.

Let ${\Sigma }_1$ and ${\Sigma }_2$ be two fixed conics in $\mathbf{P}$.

(1) It is known (see, e.g., [7]) that ${\Sigma }_1$ and ${\Sigma }_2$ have a common self-polar triangle $\Delta =\left\{D_1,D_2,D_3\right\}$. For every point $P\in \mathbf{P}\setminus \Delta$, let $p_1$ and $p_2$ be the polars of P with respect to ${\Sigma }_1$ and ${\Sigma }_2$, respectively. Then, the mapping

$$\mathcal{S}:\mathbf{P}\setminus \Delta \to \mathbf{P},P\mapsto P^{\prime }:=p_1\cap p_2$$

is said to be the Steiner correspondence with respect to ${\Sigma }_1$ and ${\Sigma }_2$. If $P^{\prime }=\mathcal{S}\left(P\right)$, then we say that the points P and $P^{\prime }$ are in Steiner correspondence. Notice that if P is a vertex of $\Delta$, e.g., $P=D_1$, then $p_1=p_2=\overleftrightarrow{D_2{D}_3}$, so the Steiner correspondence cannot be defined. Over

$$\mathbf{P}\setminus \left\{\overleftrightarrow{D_1{D}_2}\cup \overleftrightarrow{D_2{D}_3}\cup \overleftrightarrow{D_3{D}_1}\right\}$$

$\mathcal{S}$ is involutive and hence invertible.

(2) Let $l\subset \mathbf{P}$ be a line, not passing through any vertices of $\Delta$. We show that the set

$$\mathcal{S}\left(l\right)=\left\{\mathcal{S}\left(P\right)\in \mathbf{P}|P\in l\right\}$$

is a conic. Let $L_1$ and $L_2$ be the poles of l with respect to ${\Sigma }_1$ and ${\Sigma }_2$, and consider the pencils ${\mathcal{L}}_i$ with centers $L_i$, $i\in \left\{1,2\right\}$. Then, for each point $P\in l$, the point $P^{\prime }=\mathcal{S}\left(P\right)$ can be obtained as the intersection of two corresponding lines in ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$. Since there is a projectivity between ${\mathcal{L}}_i$ and the range of all points on l for $i\in \left\{1,2\right\}$, it follows that we also have a projectivity $f:{\mathcal{L}}_1\to {\mathcal{L}}_2$. Then, by Steiner’s characterization of conics, the locus of points $m\cap f\left(m\right),m\in {\mathcal{L}}_1$ is a conic. Clearly, this conic is just the set $\mathcal{S}\left(l\right)$.

(3) Observe that the conic $\mathcal{S}\left(l\right)$ contains the vertices of $\Delta$, since $\mathcal{S}$ sends every side line of $\Delta$ into the vertex opposite to the side. From the involutiveness of $\mathcal{S}$, it follows that the image of a conic passing through a vertex of $\Delta$ is a line.

(4) By the reasoning applied in Observation (2), we can also see that the Steiner correspondence sends the pairs of an involution of points on l to the pairs of an involution of points on the conic $\mathcal{S}\left(l\right)$.

(5) Suppose, finally, that the line $l\subset \mathbf{P}$ passes through a vertex $D\in \Delta$. We claim that, in this case, the image of $l\setminus \left\{D\right\}$ under $\mathcal{S}$ is a range of points on a line. Indeed, using the same notation as in Observation (2), the poles $L_1$ and $L_2$ are on the side line d opposite to D. d is the polar of D, so the projectivity $f:{\mathcal{L}}_1\to {\mathcal{L}}_2$ sends d into itself. Therefore, f is a perspectivity, and the points $m\cap f\left(m\right)$ and $(m\in {\mathcal{L}}_1)$ are colinear. Again, this point set is just $\mathcal{S}(l\setminus \left\{D\right\})$.

Now, we are in a position to prove Theorem 2 in the given Pappian plane $\mathbf{P}$. Consider two conics ${\Sigma }_1$ and ${\Sigma }_2$ with the same self-polar triangle $DEF$. Let

$$\mathcal{S}:\mathbf{P}\setminus \left\{D,E,F\right\}\to \mathbf{P},P\mapsto \mathcal{S}\left(P\right)=P^{\prime }$$

be the Steiner correspondence with respect to ${\Sigma }_1$ and ${\Sigma }_2$. Then, the sets

$$\left\{U^{\prime },V^{\prime },W^{\prime }\right\},\left\{U^{\prime },A^{\prime },C^{\prime }\right\},\left\{V^{\prime },B^{\prime },C^{\prime }\right\},\left\{W^{\prime },A^{\prime },B^{\prime }\right\}$$

are colinear. So, via Observation (5),

$$X^{\prime }=\overleftrightarrow{D{A}^{\prime }}\cap \overleftrightarrow{U^{\prime }{V}^{\prime }},Y^{\prime }=\overleftrightarrow{D{B}^{\prime }}\cap \overleftrightarrow{U^{\prime }{V}^{\prime }},Z^{\prime }=\overleftrightarrow{D{C}^{\prime }}\cap \overleftrightarrow{U^{\prime }{V}^{\prime }}.$$

Thus, the opposite side lines of the complete quadrangle $A^{\prime }{B}^{\prime }{C}^{\prime }D$ meet the line $\overleftrightarrow{U^{\prime }{V}^{\prime }}$ at the pairs of points $\left(U^{\prime }{X}^{\prime }\right)$, $\left(V^{\prime }{Y}^{\prime }\right)$, $\left(W^{\prime }{Z}^{\prime }\right)$. These are the pairs of an involution on the line $\overleftrightarrow{U^{\prime }{V}^{\prime }}$. Therefore, in view of Observation (4), their images $\left(UX\right)$, $\left(VY\right)$, and $\left(WZ\right)$ under $\mathcal{S}$ are the pairs of an involution on the conic $\Sigma$. According to our construction, this is equivalent to the property that the lines $\overleftrightarrow{UX}$, $\overleftrightarrow{VY}$, and $\overleftrightarrow{WZ}$ are concurrent.