---
title: Central Quadrilateral in Planar Geometry
url: https://www.emergentmind.com/topics/central-quadrilateral
type: topic
---

# Central Quadrilateral in Planar Geometry

Searching arXiv for the papers and related terminology.
A central quadrilateral is, in the narrowest sense, a quadrilateral obtained by placing the same triangle center in four triangles canonically associated with a reference quadrilateral \(ABCD\), then joining the four resulting points in a prescribed cyclic order. In the current literature this notion appears in two principal forms. One uses the four half triangles cut out by the diagonals, namely \(\triangle BCD\), \(\triangle ACD\), \(\triangle ABD\), and \(\triangle ABC\); the other uses four quarter or radial triangles determined by a diagonal point or by an arbitrary interior “radiator” point. Closely related work also studies the Newton line and associated center loci of conics tangent to, or passing through, the quadrilateral, thereby extending the “central” viewpoint beyond the four-vertex construction itself [2506.17240], [2509.12219].

## 1. Definitions and principal variants

The recent arXiv literature uses the term *central quadrilateral* for several closely connected constructions. In all of them, the reference figure is a quadrilateral \(ABCD\), and a single triangle center \(X(n)\) from Kimberling’s Encyclopedia of Triangle Centers (ETC) is placed in each member of a four-triangle family.

| Variant | Four source triangles | Output quadrilateral |
|---|---|---|
| Half-triangle construction | \(\triangle BCD\), \(\triangle ACD\), \(\triangle ABD\), \(\triangle ABC\) | \(EFGH\) |
| Quarter/radial construction | \(\triangle ABE\), \(\triangle BCE\), \(\triangle CDE\), \(\triangle DAE\) or \(\triangle APB\), \(\triangle BPC\), \(\triangle CPD\), \(\triangle DPA\) | \(FGHI\) or \(Q_1Q_2Q_3Q_4\) |
| Newton-line center loci | Centers of tangent conics or passing conics associated with the four extended sides or four vertices | Not a four-vertex quadrilateral, but a central locus |

In the half-triangle construction, the diagonals \(AC\) and \(BD\) of a simple quadrilateral determine four associated triangles. For a fixed center \(X_k\), the points
\[
E=X_k(\triangle BCD),\quad F=X_k(\triangle ACD),\quad G=X_k(\triangle ABD),\quad H=X_k(\triangle ABC)
\]
form the central quadrilateral \(EFGH\), usually taken in the cyclic order \(E\to F\to G\to H\), though alternative orders such as \(GHEF\) or \(HGFE\) are used when needed to exhibit perspectivity or similarity [2506.17240].

In the quarter or radial construction, a point \(E\) or \(P\) in the plane of \(ABCD\) is first chosen. When \(E\) is the intersection of the diagonals, the four triangles are often called quarter triangles; in the more general 2025 formulation, \(E\) is an arbitrary interior “radiator,” and the triangles are \(ABE\), \(BCE\), \(CDE\), and \(DAE\). Placing the same center \(X(n)\) in each produces a central quadrilateral \(FGHI\), which need not be convex [2509.12219], [2209.06008].

A broader associated usage centers on the Newton line. For a general quadrilateral \(Q\), the line through the midpoints of the diagonals is the Newton line. The centers of all ellipses and hyperbolas tangent to the four extended sides lie on this line, and the centers of all conics through the four vertices lie on a nine-point conic. This does not itself define a four-vertex central quadrilateral, but it places the Newton line at the center of the same quadrilateral-center program [2211.09764]. The 2017 work on the “two incenters” of a convex quadrilateral similarly identifies two canonical opposite-side tangent-circle centers \(I_1\) and \(I_2\) on the Newton line [1712.02207].

## 2. Coordinate models, triangle centers, and symbolic representation

The computational and analytic framework is based on trilinear, barycentric, and Cartesian coordinates. In the half-triangle setting, if a point has trilinear coordinates \(t_a:t_b:t_c\) in a triangle \(XYZ\) with opposite side lengths \(a,b,c\), then its barycentric coordinates are \((a t_a:b t_b:c t_c)\), and the Cartesian point is
\[
P=\frac{\lambda X+\mu Y+\nu Z}{\lambda+\mu+\nu},\qquad (\lambda,\mu,\nu)=(a t_a,b t_b,c t_c).
\]
This conversion is applied separately in each half triangle, so \(E\) is computed in \(\triangle BCD\), \(F\) in \(\triangle ACD\), and so on [2506.17240].

The same conversion governs the radial setting. If \(X(n)\) has trilinears \((\alpha:\beta:\gamma)\) in \(\triangle PQR\), with \(a=|QR|\), \(b=|RP|\), \(c=|PQ|\), then the barycentrics are \((\alpha a:\beta b:\gamma c)\), and
\[
X=\frac{\alpha a\,P+\beta b\,Q+\gamma c\,R}{\alpha a+\beta b+\gamma c}.
\]
This is the formula used in the analytic proofs of the radiator-based paper [2509.12219].

The half-triangle study gives explicit formulas for several classical centers. For a triangle with angles \(A,B,C\) and sides \(a=BC\), \(b=CA\), \(c=AB\), it records:
\[
X_1\text{ incenter}: [1:1:1],\quad (a:b:c);
\]
\[
X_2\text{ centroid}: [1/a:1/b:1/c],\quad (1:1:1);
\]
\[
X_4\text{ circumcenter}: [\cos A:\cos B:\cos C],\quad (a\cos A:b\cos B:c\cos C);
\]
\[
X_3\text{ orthocenter}: [\sec A:\sec B:\sec C],\quad (a\sec A:b\sec B:c\sec C);
\]
\[
X_6\text{ symmedian point}: [a:b:c],\quad (a^2:b^2:c^2).
\]
The nine-point center \(X_5\) is likewise given by trilinears \([\cos(B-C):\cos(C-A):\cos(A-B)]\) and the corresponding barycentrics [2506.17240].

The papers are systematic in scope. The half-triangle study places \(X_n\) for \(n=1,\dots,1000\), excluding points at infinity, across 28 quadrilateral families; quadrilateral centers are additionally drawn from the Encyclopedia of Quadri-Figures (EQF) when cyclic or tangential structures are involved [2506.17240]. The radiator-based shape study also tests \(1\le n\le 1000\), again using ETC data, but now across six radiator choices including an arbitrary point, the diagonal point, the Steiner point, the Poncelet point, the vertex centroid, and the area centroid [2509.12219].

For exact proofs, the half-triangle paper adopts a barycentric model with \(\triangle ABC\) as reference and \(D=(p:q:r)\), with class constraints encoded as polynomial identities in \(a,b,c,p,q,r\). Mathematica, together with the package `baricentricas.m`, is used for exact symbolic verification of equalities and incidences [2506.17240]. Earlier work likewise relies on normalized barycentrics, determinant area formulas, and change-of-coordinate formulas between nested triangles [2209.06008].

## 3. Half-triangle central quadrilaterals and comparison with the reference quadrilateral

The half-triangle program asks how the central quadrilateral \(EFGH\) compares with the reference quadrilateral \(ABCD\). The 2025 paper treats congruence, similarity, equal area, rational area ratios, equal perimeter, equality of circumcircles, equality of quadrilateral centers, perspectivity, homothety, and common circumconics [2506.17240].

For a general convex quadrilateral, the most prominent theorem concerns the centroid construction. If \(E,F,G,H\) are the centroids of the half triangles, then \(ABCD\) and \(EFGH\) are similar with ratio of similitude \(3\); equivalently, \(EFGH\) is the image of \(ABCD\) under the affine map
\[
X\mapsto m+\frac{X-m}{3},
\]
where \(m=(A+B+C+D)/4\) is the quadrilateral centroid. Hence \([ABCD]=9[EFGH]\), the two quadrilaterals are homothetic, and their centroids coincide [2506.17240]. The same phenomenon already appears in the 2022 shape paper, where the centroid-based half-triangle central quadrilateral is stated to be similar to \(ABCD\) with ratio \(3\) [2205.00870].

A second general theorem concerns the circumcenter construction in the 2025 half-triangle notation. With half-triangle circumcenters, \([ABCD]=[EFGH]\), and the two quadrilaterals share a common circumconic that is a rectangular hyperbola centered at the Euler–Poncelet point \(QA\text{-}P2\); in particular,
\[
\operatorname{ponce}[ABCD]=\operatorname{ponce}[EFGH].
\]
The same paper further proves
\[
\operatorname{ponce}[EFGH]=\operatorname{stein}[ABCD]
\]
for the orthocenter construction, and
\[
m[EFGH]=\operatorname{ponce}[ABCD]
\]
for the nine-point-center construction [2506.17240].

Special quadrilateral classes produce stronger statements. In a tangential quadrilateral, using incenters in the half triangles makes \(ABCD\) and \(GHEF\) perspective with perspector equal to the incenter \(I\); explicitly, the lines \(AG\), \(BH\), \(CE\), and \(DF\) concur at \(I\) [2506.17240]. In a cyclic quadrilateral, any center \(X\) lying on the circumcircle of a triangle yields half-triangle points \(E,F,G,H\) all lying on the circumcircle of \(ABCD\), so the reference and central quadrilaterals have the same circumcircle. For the specific center \(X399\), the circumcircles are concentric with radii in ratio \(1:2\) [2506.17240].

Several families recur. In cyclic quadrilaterals, homotheties arise for centers on the Euler line with constant Shinagawa coefficients, including \(X2\), \(X4\), \(X5\), \(X20\), \(X140\), \(X376\), \(X381\), \(X382\), \(X546\)–\(X550\), \(X631\), and \(X632\) [2506.17240]. In kites, the central quadrilateral is itself a kite, with \(EF=EH\) and \(GF=GH\), and the paper proves that the Steiner point of \(EFGH\) is the midpoint of \(EG\) [2506.17240]. In rectangles, the central quadrilateral is again a rectangle, a theorem recorded in the authors’ earlier work and referenced in the 2025 paper [2506.17240].

Older half-triangle results remain important as part of the taxonomy. For cyclic quadrilaterals, half-triangle incenters yield a rectangle, a result described as the “Japanese rectangle”; for general quadrilaterals, half-triangle orthocenters yield equal area; for orthodiagonal quadrilaterals, several Euler-line families yield orthodiagonal half-triangle central quadrilaterals [2205.00870]. The later papers reinterpret many such phenomena through center coincidences such as \(m\), \(\operatorname{ponce}\), \(\operatorname{stein}\), and \(dp\).

## 4. Radial and quarter-triangle constructions: shape theorems

The radial construction starts from a point in the plane of the quadrilateral rather than from the diagonals alone. The 2025 paper studies six radiator choices and focuses on the shape of the resulting central quadrilateral \(FGHI\): parallelogram, rhombus, rectangle, square, trapezoid, kite, cyclic, tangential, orthodiagonal, and compound classes such as bicentric or equidiagonal orthodiagonal [2509.12219].

The universal result for an arbitrary radiator \(E\) is the centroid theorem. If the same centroid \(X(2)\) is placed in the four radial triangles, then \(FGHI\) is a parallelogram whose sides are parallel to the diagonals \(AC\) and \(BD\) of \(ABCD\). In equidiagonal quadrilaterals this parallelogram is a rhombus; in orthodiagonal quadrilaterals it is a rectangle; and when \(ABCD\) is both equidiagonal and orthodiagonal, it is a square [2509.12219]. The earlier 2022 comparison paper gives the exact area relation for the same arbitrary-radiator centroid construction:
\[
[ABCD]=\frac{9}{2}[FGHI].
\]
It also writes the four vertices explicitly as
\[
Q_1=\frac{A+B+P}{3},\quad Q_2=\frac{B+C+P}{3},\quad Q_3=\frac{C+D+P}{3},\quad Q_4=\frac{D+A+P}{3},
\]
from which the side directions
\[
Q_2-Q_1=\frac{C-A}{3},\qquad Q_3-Q_2=\frac{D-B}{3}
\]
follow immediately [2209.06008].

When the radiator is constrained by symmetry, there are family-wide shape statements valid for every triangle center. If \(ABCD\) is a kite with \(AB=AD\) and the radiator lies on \(AC\), then \(FGHI\) is an isosceles trapezoid for any \(X(n)\). If \(ABCD\) is an isosceles trapezoid with \(AD\parallel BC\) and \(AB=CD\), and the radiator lies on the perpendicular bisector of \(BC\), then \(FGHI\) is a kite for any \(X(n)\) [2509.12219]. These are radiator-based analogues of the older diagonal-point quarter-triangle theorems stating that kites yield isosceles trapezoids and isosceles trapezoids yield kites for all centers [2205.00870].

The Steiner-point and Poncelet-point radiators generate richer class-dependent phenomena. For cyclic \(ABCD\), with the Steiner point as radiator, the paper proves that \(X(3)\), \(X(399)\), and all centers in a set \(T\) give tangential \(FGHI\) with incenter equal to the radiator; for \(X(3)\), the inradius equals half the circumradius of \(ABCD\), while for centers in \(T\) the incircle of \(FGHI\) coincides with the circumcircle of \(ABCD\) [2509.12219]. For orthodiagonal \(ABCD\), the same radiator gives cyclic \(FGHI\) for \(X(3)\) and a trapezoid \(FHIG\) with \(FH\parallel GI\) for \(X(4)\) [2509.12219]. In Hjelmslev quadrilaterals, taking the Poncelet point as radiator yields a trapezoid with \(FG\parallel HI\) for \(X(3)\), and a tangential trapezoid with incenter equal to the radiator and \(FI\parallel GH\) for \(X(4)\) [2509.12219].

The vertex centroid as radiator supports one of the most specific 2025 discoveries. If \(ABCD\) is equidiagonal and the radial center is \(X(591)\), then \(FGHI\) is orthodiagonal; the paper records symbolic and numerical verification of
\[
(H-F)\cdot(I-G)=0.
\]
In equidiagonal orthodiagonal quadrilaterals, the same radiator gives a parallelogram for \(X(491)\) and \(X(615)\) [2509.12219].

Earlier quarter-triangle work identifies broad center-function families. For general quadrilaterals with the diagonal point as radiator, Euler-line family centers with center function \(\cos B\cos C+k\cos A\) always yield a parallelogram; incenter constructions are always orthodiagonal; in equidiagonal quadrilaterals, Nagel-line family centers \((b+c)/a+k\) yield orthodiagonal central quadrilaterals with diagonals parallel to the bimedians; and in orthodiagonal quadrilaterals, the symmedian \(X6\) yields a cyclic central quadrilateral [2205.00870]. These family-level results remain structurally important because they organize many individual ETC indices into coherent geometric behaviors.

## 5. Newton line, tangent conics, and the broader central axis

A second branch of the literature shifts attention from four triangle centers to loci of conic centers naturally associated with a quadrilateral. Its basic object is the Newton line, the line through the midpoints
\[
M_{AC}=\frac{A+C}{2},\qquad M_{BD}=\frac{B+D}{2}
\]
of the two diagonals. In parametric form,
\[
N(t)=M_{AC}+t(M_{BD}-M_{AC}),\qquad t\in\mathbb{R}.
\]
For nontrapezoidal quadrilaterals, the midpoint of the segment joining the intersections of opposite extended sides also lies on this line [2211.09764].

The 2022 paper generalizes Newton’s quadrilateral theorem from circles to all centered nondegenerate conics tangent to the four extended sides. If \(H\) is an ellipse or hyperbola tangent to the four extended side-lines of a non-parallelogram quadrilateral \(Q\), then its center \(O\) lies on the Newton line [2211.09764]. In centered form
\[
(x-O)^\top B(x-O)=1,
\]
tangency of a line \(L:n^\top x=b\) is characterized exactly by
\[
n^\top B^{-1}n=(b-n^\top O)^2.
\]
Using a pencil of quadratic forms
\[
\mathcal{C}_\lambda(x)=\sum_{i=1}^4 \lambda_i \ell_i(x)^2,
\]
the paper derives
\[
A(\lambda)=\sum \lambda_i n_i n_i^\top,\qquad 
b(\lambda)=-2\sum \lambda_i b_i n_i,\qquad
O(\lambda)=A(\lambda)^{-1}\sum \lambda_i b_i n_i,
\]
and proves that \(O(\lambda)\) traces precisely the Newton line in the nondegenerate region [2211.09764].

The converse is nearly as strong. Every point on the Newton line, except three singular points, is the center of some ellipse or hyperbola tangent to the four extended sides. The excluded points are the two diagonal midpoints and the midpoint \(T\) of the two opposite-side intersection points \(P=L_{AB}\cap L_{CD}\) and \(Q=L_{BC}\cap L_{DA}\) [2211.09764]. In the same paper, Minthorn’s theorem is presented in affine form: the centers of all conics through the four vertices of \(Q\) lie on a nine-point conic \(H\), whose center is
\[
\frac{A+B+C+D}{4},
\]
and \(H\) is a hyperbola if and only if \(Q\) is strictly convex [2211.09764].

The 2017 “two incenters” paper gives a metric version of the same central-axis phenomenon. For any convex quadrilateral \(ABCD\), there is a unique point \(I_1\) on the Newton line equidistant from the opposite sides \(AB\) and \(CD\), and a unique point \(I_2\) on the Newton line equidistant from \(BC\) and \(AD\). If their corresponding radii are \(r_1\) and \(r_2\), then the harmonic mean
\[
r=\frac{2r_1r_2}{r_1+r_2}
\]
satisfies the universal area–perimeter identity
\[
A=\frac{pr}{2},
\]
where \(A\) is the area and \(p\) the perimeter of \(ABCD\) [1712.02207]. In the tangential case, \(I_1\equiv I_2\) and \(r_1=r_2=r\), recovering the classical inradius formula. In the cyclic case, \(I_1\equiv I_2\) but generally \(r_1\ne r_2\) [1712.02207].

Taken together, these papers identify the Newton line as the principal affine axis of quadrilateral-centered geometry. The point-placing central-quadrilateral constructions and the tangent/passing-conic center loci are distinct objects, but they share the same organizing theme: local triangle or conic data aggregate into rigid global structures governed by diagonal midpoints, bimedians, and canonical quadrilateral centers.

## 6. Computational methodology, notation issues, and open problems

The central-quadrilateral literature is explicitly computational as well as synthetic. The main engine is GeometricExplorer, used to place ETC centers \(X(n)\) for \(1\le n\le 1000\) and test shape, area, perimeter, similarity, perspectivity, homothety, and conic-sharing phenomena across 28 quadrilateral classes [2506.17240], [2509.12219]. Typical floating precision is about 15 digits. Shape tests rely on vector proportionality for parallelism, dot products for perpendicularity, Euclidean norms for equal lengths, shoelace area, and determinant tests for concyclicity [2509.12219]. In the half-triangle comparison paper, congruence is checked after Procrustes alignment, and rational area ratios are fitted to rationals with denominator less than \(10\) [2506.17240]. In the earlier 2022 comparison paper, the primary computational pass uses denominators \(\le 6\), with supplementary larger denominators reported separately [2209.06008].

Symbolic verification complements the numerical search. The 2025 half-triangle paper encodes geometric class constraints, such as cyclicity, orthodiagonality, parallelogram, rectangle, and kite conditions, as algebraic identities in \(a,b,c,p,q,r\) under the barycentric model \(D=(p:q:r)\) [2506.17240]. The radiator-based shape paper likewise combines geometric proofs where available with exact barycentric computation in Mathematica [2509.12219]. Dynamic software such as Geometer’s Sketchpad and GeoGebra is used to confirm invariance under point motions [2509.12219].

A technical issue in reading this literature is notational nonuniformity. The older quarter-triangle and radiator papers use the standard ETC indexing in which \(X(3)\) is the circumcenter and \(X(4)\) the orthocenter [2209.06008], [2509.12219]. By contrast, the 2025 half-triangle paper labels \(X_3\) as the orthocenter and \(X_4\) as the circumcenter [2506.17240]. This is a matter of local paper convention, but it materially affects the interpretation of statements such as “Theorem (X3)” or “Theorem (X4).”

The papers also delineate robustness and edge cases. Points at infinity are excluded; self-intersecting quadrilaterals are excluded in the main half-triangle study; and the central quadrilateral may be nonconvex even when \(ABCD\) is convex [2506.17240]. Ordering is significant: \(E,F,G,H\) or \(F,G,H,I\) are usually taken cyclically, but alternative orderings such as \(GHEF\), \(HGFE\), or \(FHIG\) are sometimes necessary to reveal the correct perspectivity or similarity statement [2506.17240], [2509.12219].

The open problems are both structural and classificatory. In the half-triangle setting, the recent paper asks for purely geometric proofs of the \(X3\), \(X4\), and \(X5\) global theorems; a full characterization of the Euler-line centers with constant Shinagawa coefficients that yield homotheties in cyclic quadrilaterals; a complete proof of the trapezoid \(X3\) similarity/orthogonality phenomenon; the uniqueness of \(X2\) for the property \(m[ABCD]=m[EFGH]\) for all quadrilaterals; and the classification of centers for which the reference and central quadrilaterals share an inconic [2506.17240]. In the radial setting, the 2025 shape paper asks whether \(X(2)\) is the only center that yields a parallelogram for an arbitrary radiator in a general quadrilateral, whether \(X(591)\) is unique in producing orthodiagonality for equidiagonal quadrilaterals with radiator at the vertex centroid, and whether cyclic or trapezoidal outputs force the center to be \(X(3)\) or \(X(4)\) in certain configurations [2509.12219].

Across these works, the central quadrilateral has become a testbed for computer-discovered planar geometry. The repeated emergence of Varignon-type averaging, Newton-line centrality, Euler-line and Nagel-line families, and EQF center coincidences suggests that the construction is less a single theorem than a framework: one in which local triangle-center data interact with global quadrilateral symmetries in ways that are rigid enough for exact classification, yet rich enough to support large-scale computational discovery [2506.17240].

Source: https://www.emergentmind.com/topics/central-quadrilateral