ex-tangential quadrilateral

Four extensions, one circle

Adopt the usual convex definition requiring an outside circle tangent to all four side extensions, as described in [Different cases of a tangential quadrilateral theorem](https://www.tandfonline.com/doi/full/10.1080/0020739X.2025.2572374).

1. Does the instance provide one external circle tangent to all four side extensions? definition
2. Which source and definition passage justify the convention used for classification? provenance

Convexity is a declared boundary

The terminology is also used for concave objects in [Aebi and Cairns, Lattice Equable Quadrilaterals III](https://arxiv.org/abs/2111.06453). Such usage requires an attributed alternative convention rather than automatic inclusion.

1. Does the supplying source permit concave quadrilaterals, and what contact condition does it use? boundary
2. Should this candidate be assessed under the working definition or retained as an attributed alternative usage? action

Admissible vertex cycle

Record A, B, C, D in boundary order and check distinctness, nonzero side lengths, simplicity and strict convexity.

1. What coordinates or exact construction establish the boundary order A, B, C, D? measurement
2. Do any repeated vertices, collinear consecutive triples or crossing sides invalidate the working scope? boundary

Four exterior contact feet

For each side V_i V_(i+1), record its contact foot T_i and parameter t_i; an extension contact requires t_i outside [0,1].

1. What is each contact parameter, and beyond which endpoint does its contact lie? measurement
2. Does any contact lie on a side segment or at a vertex, requiring rejection under the working convention? boundary

Common distance and exterior placement

Check distance(O, line_i) = r for every side line with finite r > 0, then check that the circle is external to the polygon and that all contact parameters are admissible.

1. What center and positive radius satisfy all four line-distance equations? measurement
2. What geometric test certifies that the circle lies outside the quadrilateral? boundary

Qualified membership decision

Record whether the equalities and inequalities are proved exactly or checked approximately; a failed candidate circle alone does not disprove the existence of another.

1. Are all required equalities and strict inequalities exact, interval-certified or merely within numerical tolerance? measurement
2. If membership is rejected, does the evidence exclude every eligible circle or only the current candidate? action

Bisector branch validation

For intersecting line pairs, equal-distance loci are their angle bisectors. Intersections of selected branches supply candidates that still require the remaining tangency and extension checks.

1. Which angle-bisector branches or equivalent signed-distance equations generated each candidate center? action
2. Which candidates survive all four line-distance, contact-location and exterior-placement checks? measurement

Similarity preserves the certificate

Translations, rotations, reflections and uniform scaling by a nonzero factor preserve circular tangency and extension membership; lengths scale by the absolute scale factor.

1. Is the proposed transformation a Euclidean similarity, and how does it transform the center, radius and contact points? action
2. If vertices are moved independently or scaling is nonuniform, what fresh verification establishes ex-tangentiality? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence? provenance