Central Quadrilateral in Planar Geometry

• Central quadrilateral is a figure formed by placing the same triangle center in four triangles derived from a reference quadrilateral.
• Variants include half-triangle, quarter-triangle, and radial constructions, each employing centers like the centroid, circumcenter, or orthocenter.
• It exhibits properties such as similarity, homothety, and links to the Newton line and conic center loci, enriching planar geometric analysis.

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 (Rabinowitz et al., 5 Jun 2025, Rabinowitz et al., 2 Sep 2025).

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 BCD$0	$\triangle BCD$1
Quarter/radial construction	$\triangle BCD$2, $\triangle BCD$3, $\triangle BCD$4, $\triangle BCD$5 or $\triangle BCD$6, $\triangle BCD$7, $\triangle BCD$8, $\triangle BCD$9	$\triangle ACD$0 or $\triangle ACD$1
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 $\triangle ACD$2 and $\triangle ACD$3 of a simple quadrilateral determine four associated triangles. For a fixed center $\triangle ACD$4, the points

$\triangle ACD$5

form the central quadrilateral $\triangle ACD$6, usually taken in the cyclic order $\triangle ACD$7, though alternative orders such as $\triangle ACD$8 or $\triangle ACD$9 are used when needed to exhibit perspectivity or similarity (Rabinowitz et al., 5 Jun 2025).

In the quarter or radial construction, a point $\triangle ABD$0 or $\triangle ABD$1 in the plane of $\triangle ABD$2 is first chosen. When $\triangle ABD$3 is the intersection of the diagonals, the four triangles are often called quarter triangles; in the more general 2025 formulation, $\triangle ABD$4 is an arbitrary interior “radiator,” and the triangles are $\triangle ABD$5, $\triangle ABD$6, $\triangle ABD$7, and $\triangle ABD$8. Placing the same center $\triangle ABD$9 in each produces a central quadrilateral $\triangle ABC$0, which need not be convex (Rabinowitz et al., 2 Sep 2025, Rabinowitz et al., 2022).

A broader associated usage centers on the Newton line. For a general quadrilateral $\triangle ABC$1, 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 (Kaldybayev, 2022). The 2017 work on the “two incenters” of a convex quadrilateral similarly identifies two canonical opposite-side tangent-circle centers $\triangle ABC$2 and $\triangle ABC$3 on the Newton line (Dergiades et al., 2017).

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 $\triangle ABC$4 in a triangle $\triangle ABC$5 with opposite side lengths $\triangle ABC$6, then its barycentric coordinates are $\triangle ABC$7, and the Cartesian point is

$\triangle ABC$8

This conversion is applied separately in each half triangle, so $\triangle ABC$9 is computed in $ABCD$0, $ABCD$1 in $ABCD$2, and so on (Rabinowitz et al., 5 Jun 2025).

The same conversion governs the radial setting. If $ABCD$3 has trilinears $ABCD$4 in $ABCD$5, with $ABCD$6, $ABCD$7, $ABCD$8, then the barycentrics are $ABCD$9, and

$X(n)$0

This is the formula used in the analytic proofs of the radiator-based paper (Rabinowitz et al., 2 Sep 2025).

The half-triangle study gives explicit formulas for several classical centers. For a triangle with angles $X(n)$1 and sides $X(n)$2, $X(n)$3, $X(n)$4, it records: $X(n)$5

$X(n)$6

$X(n)$7

$X(n)$8

$X(n)$9

The nine-point center $\triangle BCD$0 is likewise given by trilinears $\triangle BCD$1 and the corresponding barycentrics (Rabinowitz et al., 5 Jun 2025).

The papers are systematic in scope. The half-triangle study places $\triangle BCD$2 for $\triangle BCD$3, 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 (Rabinowitz et al., 5 Jun 2025). The radiator-based shape study also tests $\triangle BCD$4, 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 (Rabinowitz et al., 2 Sep 2025).

For exact proofs, the half-triangle paper adopts a barycentric model with $\triangle BCD$5 as reference and $\triangle BCD$6, with class constraints encoded as polynomial identities in $\triangle BCD$7. Mathematica, together with the package baricentricas.m, is used for exact symbolic verification of equalities and incidences (Rabinowitz et al., 5 Jun 2025). Earlier work likewise relies on normalized barycentrics, determinant area formulas, and change-of-coordinate formulas between nested triangles (Rabinowitz et al., 2022).

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

The half-triangle program asks how the central quadrilateral $\triangle BCD$8 compares with the reference quadrilateral $\triangle BCD$9. 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 (Rabinowitz et al., 5 Jun 2025).

For a general convex quadrilateral, the most prominent theorem concerns the centroid construction. If $\triangle ACD$0 are the centroids of the half triangles, then $\triangle ACD$1 and $\triangle ACD$2 are similar with ratio of similitude $\triangle ACD$3; equivalently, $\triangle ACD$4 is the image of $\triangle ACD$5 under the affine map

$\triangle ACD$6

where $\triangle ACD$7 is the quadrilateral centroid. Hence $\triangle ACD$8, the two quadrilaterals are homothetic, and their centroids coincide (Rabinowitz et al., 5 Jun 2025). The same phenomenon already appears in the 2022 shape paper, where the centroid-based half-triangle central quadrilateral is stated to be similar to $\triangle ACD$9 with ratio $\triangle ABD$0 (Rabinowitz et al., 2022).

A second general theorem concerns the circumcenter construction in the 2025 half-triangle notation. With half-triangle circumcenters, $\triangle ABD$1, and the two quadrilaterals share a common circumconic that is a rectangular hyperbola centered at the Euler–Poncelet point $\triangle ABD$2; in particular,

$\triangle ABD$3

The same paper further proves

$\triangle ABD$4

for the orthocenter construction, and

$\triangle ABD$5

for the nine-point-center construction (Rabinowitz et al., 5 Jun 2025).

Special quadrilateral classes produce stronger statements. In a tangential quadrilateral, using incenters in the half triangles makes $\triangle ABD$6 and $\triangle ABD$7 perspective with perspector equal to the incenter $\triangle ABD$8; explicitly, the lines $\triangle ABD$9, $\triangle BCD$00, $\triangle BCD$01, and $\triangle BCD$02 concur at $\triangle BCD$03 (Rabinowitz et al., 5 Jun 2025). In a cyclic quadrilateral, any center $\triangle BCD$04 lying on the circumcircle of a triangle yields half-triangle points $\triangle BCD$05 all lying on the circumcircle of $\triangle BCD$06, so the reference and central quadrilaterals have the same circumcircle. For the specific center $\triangle BCD$07, the circumcircles are concentric with radii in ratio $\triangle BCD$08 (Rabinowitz et al., 5 Jun 2025).

Several families recur. In cyclic quadrilaterals, homotheties arise for centers on the Euler line with constant Shinagawa coefficients, including $\triangle BCD$09, $\triangle BCD$10, $\triangle BCD$11, $\triangle BCD$12, $\triangle BCD$13, $\triangle BCD$14, $\triangle BCD$15, $\triangle BCD$16, $\triangle BCD$17–$\triangle BCD$18, $\triangle BCD$19, and $\triangle BCD$20 (Rabinowitz et al., 5 Jun 2025). In kites, the central quadrilateral is itself a kite, with $\triangle BCD$21 and $\triangle BCD$22, and the paper proves that the Steiner point of $\triangle BCD$23 is the midpoint of $\triangle BCD$24 (Rabinowitz et al., 5 Jun 2025). In rectangles, the central quadrilateral is again a rectangle, a theorem recorded in the authors’ earlier work and referenced in the 2025 paper (Rabinowitz et al., 5 Jun 2025).

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 (Rabinowitz et al., 2022). The later papers reinterpret many such phenomena through center coincidences such as $\triangle BCD$25, $\triangle BCD$26, $\triangle BCD$27, and $\triangle BCD$28.

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 $\triangle BCD$29: parallelogram, rhombus, rectangle, square, trapezoid, kite, cyclic, tangential, orthodiagonal, and compound classes such as bicentric or equidiagonal orthodiagonal (Rabinowitz et al., 2 Sep 2025).

The universal result for an arbitrary radiator $\triangle BCD$30 is the centroid theorem. If the same centroid $\triangle BCD$31 is placed in the four radial triangles, then $\triangle BCD$32 is a parallelogram whose sides are parallel to the diagonals $\triangle BCD$33 and $\triangle BCD$34 of $\triangle BCD$35. In equidiagonal quadrilaterals this parallelogram is a rhombus; in orthodiagonal quadrilaterals it is a rectangle; and when $\triangle BCD$36 is both equidiagonal and orthodiagonal, it is a square (Rabinowitz et al., 2 Sep 2025). The earlier 2022 comparison paper gives the exact area relation for the same arbitrary-radiator centroid construction: $\triangle BCD$37 It also writes the four vertices explicitly as

$\triangle BCD$38

from which the side directions

$\triangle BCD$39

follow immediately (Rabinowitz et al., 2022).

When the radiator is constrained by symmetry, there are family-wide shape statements valid for every triangle center. If $\triangle BCD$40 is a kite with $\triangle BCD$41 and the radiator lies on $\triangle BCD$42, then $\triangle BCD$43 is an isosceles trapezoid for any $\triangle BCD$44. If $\triangle BCD$45 is an isosceles trapezoid with $\triangle BCD$46 and $\triangle BCD$47, and the radiator lies on the perpendicular bisector of $\triangle BCD$48, then $\triangle BCD$49 is a kite for any $\triangle BCD$50 (Rabinowitz et al., 2 Sep 2025). 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 (Rabinowitz et al., 2022).

The Steiner-point and Poncelet-point radiators generate richer class-dependent phenomena. For cyclic $\triangle BCD$51, with the Steiner point as radiator, the paper proves that $\triangle BCD$52, $\triangle BCD$53, and all centers in a set $\triangle BCD$54 give tangential $\triangle BCD$55 with incenter equal to the radiator; for $\triangle BCD$56, the inradius equals half the circumradius of $\triangle BCD$57, while for centers in $\triangle BCD$58 the incircle of $\triangle BCD$59 coincides with the circumcircle of $\triangle BCD$60 (Rabinowitz et al., 2 Sep 2025). For orthodiagonal $\triangle BCD$61, the same radiator gives cyclic $\triangle BCD$62 for $\triangle BCD$63 and a trapezoid $\triangle BCD$64 with $\triangle BCD$65 for $\triangle BCD$66 (Rabinowitz et al., 2 Sep 2025). In Hjelmslev quadrilaterals, taking the Poncelet point as radiator yields a trapezoid with $\triangle BCD$67 for $\triangle BCD$68, and a tangential trapezoid with incenter equal to the radiator and $\triangle BCD$69 for $\triangle BCD$70 (Rabinowitz et al., 2 Sep 2025).

The vertex centroid as radiator supports one of the most specific 2025 discoveries. If $\triangle BCD$71 is equidiagonal and the radial center is $\triangle BCD$72, then $\triangle BCD$73 is orthodiagonal; the paper records symbolic and numerical verification of

$\triangle BCD$74

In equidiagonal orthodiagonal quadrilaterals, the same radiator gives a parallelogram for $\triangle BCD$75 and $\triangle BCD$76 (Rabinowitz et al., 2 Sep 2025).

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 $\triangle BCD$77 always yield a parallelogram; incenter constructions are always orthodiagonal; in equidiagonal quadrilaterals, Nagel-line family centers $\triangle BCD$78 yield orthodiagonal central quadrilaterals with diagonals parallel to the bimedians; and in orthodiagonal quadrilaterals, the symmedian $\triangle BCD$79 yields a cyclic central quadrilateral (Rabinowitz et al., 2022). 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

$\triangle BCD$80

of the two diagonals. In parametric form,

$\triangle BCD$81

For nontrapezoidal quadrilaterals, the midpoint of the segment joining the intersections of opposite extended sides also lies on this line (Kaldybayev, 2022).

The 2022 paper generalizes Newton’s quadrilateral theorem from circles to all centered nondegenerate conics tangent to the four extended sides. If $\triangle BCD$82 is an ellipse or hyperbola tangent to the four extended side-lines of a non-parallelogram quadrilateral $\triangle BCD$83, then its center $\triangle BCD$84 lies on the Newton line (Kaldybayev, 2022). In centered form

$\triangle BCD$85

tangency of a line $\triangle BCD$86 is characterized exactly by

$\triangle BCD$87

Using a pencil of quadratic forms

$\triangle BCD$88

the paper derives

$\triangle BCD$89

and proves that $\triangle BCD$90 traces precisely the Newton line in the nondegenerate region (Kaldybayev, 2022).

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 $\triangle BCD$91 of the two opposite-side intersection points $\triangle BCD$92 and $\triangle BCD$93 (Kaldybayev, 2022). In the same paper, Minthorn’s theorem is presented in affine form: the centers of all conics through the four vertices of $\triangle BCD$94 lie on a nine-point conic $\triangle BCD$95, whose center is

$\triangle BCD$96

and $\triangle BCD$97 is a hyperbola if and only if $\triangle BCD$98 is strictly convex (Kaldybayev, 2022).

The 2017 “two incenters” paper gives a metric version of the same central-axis phenomenon. For any convex quadrilateral $\triangle BCD$99, there is a unique point $\triangle ACD$00 on the Newton line equidistant from the opposite sides $\triangle ACD$01 and $\triangle ACD$02, and a unique point $\triangle ACD$03 on the Newton line equidistant from $\triangle ACD$04 and $\triangle ACD$05. If their corresponding radii are $\triangle ACD$06 and $\triangle ACD$07, then the harmonic mean

$\triangle ACD$08

satisfies the universal area–perimeter identity

$\triangle ACD$09

where $\triangle ACD$10 is the area and $\triangle ACD$11 the perimeter of $\triangle ACD$12 (Dergiades et al., 2017). In the tangential case, $\triangle ACD$13 and $\triangle ACD$14, recovering the classical inradius formula. In the cyclic case, $\triangle ACD$15 but generally $\triangle ACD$16 (Dergiades et al., 2017).

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 $\triangle ACD$17 for $\triangle ACD$18 and test shape, area, perimeter, similarity, perspectivity, homothety, and conic-sharing phenomena across 28 quadrilateral classes (Rabinowitz et al., 5 Jun 2025, Rabinowitz et al., 2 Sep 2025). 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 (Rabinowitz et al., 2 Sep 2025). In the half-triangle comparison paper, congruence is checked after Procrustes alignment, and rational area ratios are fitted to rationals with denominator less than $\triangle ACD$19 (Rabinowitz et al., 5 Jun 2025). In the earlier 2022 comparison paper, the primary computational pass uses denominators $\triangle ACD$20, with supplementary larger denominators reported separately (Rabinowitz et al., 2022).

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 $\triangle ACD$21 under the barycentric model $\triangle ACD$22 (Rabinowitz et al., 5 Jun 2025). The radiator-based shape paper likewise combines geometric proofs where available with exact barycentric computation in Mathematica (Rabinowitz et al., 2 Sep 2025). Dynamic software such as Geometer’s Sketchpad and GeoGebra is used to confirm invariance under point motions (Rabinowitz et al., 2 Sep 2025).

A technical issue in reading this literature is notational nonuniformity. The older quarter-triangle and radiator papers use the standard ETC indexing in which $\triangle ACD$23 is the circumcenter and $\triangle ACD$24 the orthocenter (Rabinowitz et al., 2022, Rabinowitz et al., 2 Sep 2025). By contrast, the 2025 half-triangle paper labels $\triangle ACD$25 as the orthocenter and $\triangle ACD$26 as the circumcenter (Rabinowitz et al., 5 Jun 2025). 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 $\triangle ACD$27 is convex (Rabinowitz et al., 5 Jun 2025). Ordering is significant: $\triangle ACD$28 or $\triangle ACD$29 are usually taken cyclically, but alternative orderings such as $\triangle ACD$30, $\triangle ACD$31, or $\triangle ACD$32 are sometimes necessary to reveal the correct perspectivity or similarity statement (Rabinowitz et al., 5 Jun 2025, Rabinowitz et al., 2 Sep 2025).

The open problems are both structural and classificatory. In the half-triangle setting, the paper asks for purely geometric proofs of the $\triangle ACD$33, $\triangle ACD$34, and $\triangle ACD$35 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 $\triangle ACD$36 similarity/orthogonality phenomenon; the uniqueness of $\triangle ACD$37 for the property $\triangle ACD$38 for all quadrilaterals; and the classification of centers for which the reference and central quadrilaterals share an inconic (Rabinowitz et al., 5 Jun 2025). In the radial setting, the 2025 shape paper asks whether $\triangle ACD$39 is the only center that yields a parallelogram for an arbitrary radiator in a general quadrilateral, whether $\triangle ACD$40 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 $\triangle ACD$41 or $\triangle ACD$42 in certain configurations (Rabinowitz et al., 2 Sep 2025).

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 (Rabinowitz et al., 5 Jun 2025).