Publications

Research Agenda for the Two-Circle Tangent Theorem: From Analytic Proof to Synthetic Understanding

We outline a research program to achieve a complete synthetic proof of the two-circle tangent theorem, formalize it in Lean, explore generalizations, and connect it to broader geometric theories such as Poncelet porisms and radical axis theory.

Contributions to the Two‑Circle Tangent Theorem: A Personal Summary

We summarize the author's contributions to the research on the two‑circle tangent theorem: a simplified analytic proof for orthogonal circles, formalization challenges, investigation of a three‑dimensional analogue, and critical review of synthetic attempts, including identification of a false collinearity assumption in inversion‑based proofs.

Does the Two‑Circle Tangent Theorem Generalize to Three Dimensions?

We investigate whether the tangent line theorem for two intersecting circles admits a natural generalization to three dimensions, where the circles become spheres intersecting in a circle. Numerical experiments indicate that the property fails in general, except when the configuration lies in a plane, suggesting the theorem is essentially planar.

The Two-Circle Tangent Theorem: A Review of Progress and Pitfalls

We review the research on the two-circle tangent theorem, summarizing established results (analytic proof, rational identity, converse theorem, orthogonal-case lemmas) and highlighting pitfalls in synthetic proof attempts, including a false collinearity assumption in inversion approaches. We outline open problems and future directions.

Power of a Point and Radical Axis Relations in the Two-Circle Tangent Theorem

We compute the powers of the orthocenter H with respect to the various circles appearing in the two-circle tangent theorem. We discover identities linking these powers to the squared distance HT² and to the rational certificate ρ². These relations suggest a synthetic proof strategy based on radical axes and power of a point.

Factorization of the Tangency Condition and a Simplified Proof of the Two-Circle Theorem

We show that the polynomial condition for the line through H parallel to AP to be tangent to the circumcircle of triangle BEF factors as (PA² - PC²) times a nonzero rational function. This factorization yields a short algebraic proof and explains why the tangency characterizes the circumcenter P.

A Synthetic Proof that Orthogonal Circles Imply a Right-Angled Triangle via Inversion

We give a short synthetic proof that when two circles intersect orthogonally, triangle BEF in the configuration of the tangent line theorem is right-angled at B. The proof uses inversion to transform the circles into perpendicular lines, preserving angles.

On a Collinearity Property in the Inverted Configuration of the Two-Circle Tangent Theorem

We present numerical evidence that points A, H', Q' are not collinear in the inverted configuration, challenging an assumption in some synthetic proof outlines. The observation highlights a subtlety in the geometry and calls for a revision of inversion-based approaches.

A Radical Axis Approach to the Two-Circle Tangent Theorem

We outline a synthetic proof strategy based on radical axes and power of points. We show that the tangency condition is equivalent to a relation among the powers of H with respect to three circles: Ω, Γ, and the circle with diameter CD.

Formalizing the Two-Circle Tangent Theorem in Lean: Challenges and Strategies

We discuss the challenges of formalizing the two-circle tangent theorem in the Lean theorem prover, focusing on algebraic complexity, square-root elimination, and geometric side conditions. We propose strategies to overcome these difficulties and illustrate them with a sketch of the orthogonal case.

The Diameter Property for Orthogonal Intersecting Circles and Its Implications for the Tangent Theorem

We prove that when two circles intersect orthogonally, the segment EF is a diameter of the circumcircle of triangle BEF. This diameter property simplifies the configuration and provides a geometric explanation for the simplification of the algebraic certificate. We discuss how this observation could lead to a synthetic proof.

Research Report on the Two-Circle Tangent Theorem: Advances and Open Problems

We summarize the research conducted on a geometric theorem involving two intersecting circles and a tangent line property, detailing the analytic proof, converse results, orthogonal-circle lemma, inversion approaches, and remaining open problems.

A Simplified Analytic Proof of the Tangent Line Theorem for Orthogonal Intersecting Circles

We show that when two circles intersect orthogonally, the coordinates of the points appearing in the tangent line theorem simplify dramatically. Using the general formulas from the existing analytic proof, we derive explicit expressions for the circumcenter O of triangle BEF and the orthocenter H, and verify that the line through H parallel to AP is tangent to the circumcircle of BEF. The computations are much simpler than in the general case, providing a clear illustration of the geometric fact.

Towards a Synthetic Proof of the Two-Circle Tangent Theorem via Inversion and Orthogonal Circles

We outline a synthetic proof strategy combining inversion at an intersection point with a lemma about orthogonal circles, reducing the theorem to a power-of-a-point relation. The approach provides geometric insight and suggests a path to a fully synthetic proof.

A Characterization of the Circumcenter via a Tangent Property in a Two-Circle Configuration

We show that in the configuration of two intersecting circles, the point P on the perpendicular bisector of CD for which the line through the orthocenter of triangle PMN parallel to AP is tangent to the circumcircle of triangle BEF is precisely the circumcenter of triangle ACD. This provides a converse to the known theorem.

Triangle Interpretation of the Two-Circle Tangent Theorem

We show that the squared distance HT² from the orthocenter H to the tangency point can be expressed elegantly in terms of the side lengths of triangle AMN, where A is an intersection point of the two circles, and M, N are their centers.

A Lemma on Orthogonal Circles in the Two-Circle Tangent Configuration

We prove that when two intersecting circles are orthogonal, the circumcenter of triangle BEF is the midpoint of segment EF, where points are defined as in the tangent line theorem. This lemma simplifies the proof of the main theorem for orthogonal circles and provides geometric insight.

Towards a Synthetic Proof via Inversion: A Geometric Interpretation of the Two-Circle Tangent Theorem

We sketch a synthetic proof of the tangent line theorem using inversion. By reducing the problem to a tangency between two circles in the inverted plane, we provide geometric interpretations of the key algebraic identity and indicate how a complete synthetic argument might be completed.

A Comprehensive Solution to a Tangent Theorem for Two Intersecting Circles

We present a complete solution to a geometric theorem involving two intersecting circles and associated points. The solution includes an analytic coordinate proof, a compact rational identity, an inversion-based geometric reduction, and an analysis of limit cases. We also discuss open problems and possible synthetic approaches.

Limit Cases and Algebraic Identities for a Tangent Theorem of Two Circles

We study the behavior of a geometric theorem about two intersecting circles as the circles become tangent or disjoint. The theorem admits a simple rational identity, which we show remains valid even in degenerate configurations. Numerical experiments support the conjecture that the tangency property holds in the limit.

Inversion and the Tangency of a Line to a Circle in a Two-Circle Configuration

We provide a detailed analysis of the configuration under inversion centered at one intersection point. We derive explicit relationships among the images of the key points and reduce the tangency condition to a simple property involving cross ratios and orthogonal circles.

Generalizations and Open Problems Related to a Two-Circle Tangent Theorem

We discuss possible extensions of the theorem to other configurations, such as circles intersecting at right angles, non‑intersecting circles, and higher‑dimensional analogues.

An Inversion Approach to the Two-Circle Tangent Theorem

We use inversion about A to transform the configuration into a simpler one where the two original circles become lines. We then reduce the tangency condition to a property of the inverted figure.

A Simple Rational Identity Underlying a Two-Circle Tangent Theorem

We present a compact rational expression for the squared radius of the circumcircle of triangle BEF and show that it equals the squared distance from its center to the line through H parallel to AP.

A Survey of a Geometry Theorem on Two Intersecting Circles

We review a geometric theorem concerning two intersecting circles and associated points, summarizing an analytic proof and discussing synthetic approaches.

A Survey of Approaches to a Tangent Line Problem for Two Intersecting Circles

We review the various contributions made towards proving a geometric theorem involving two intersecting circles, a circumcenter, an orthocenter, and a tangent line. The paper summarizes analytic, numeric, and synthetic methods, and identifies open questions.

An Inversion Approach to a Tangent Property of Two Intersecting Circles

We propose an inversion-based method to prove that the line through the orthocenter of a certain triangle parallel to a certain line is tangent to the circumcircle of another triangle formed by intersection points of two circles. The inversion simplifies the configuration and reduces the problem to a more manageable statement.

Experimental Verification of a Tangent Line Property in a Two-Circle Configuration

We provide strong numerical evidence that the line through H parallel to AP is tangent to the circumcircle of BEF, using extensive random testing across admissible parameters.

A Property of the Orthocenter in the Configuration of Two Intersecting Circles

We show that the orthocenter H of triangle PMN lies on the same vertical line as the circumcenter P of triangle ACD.

A Coordinate Geometry Proof of a Tangent Line Property in a Two-Circle Configuration

We prove that in the configuration of two intersecting circles with centers M and N, the line through the orthocenter of triangle PMN parallel to line AP is tangent to the circumcircle of triangle BEF, using analytic geometry and symbolic verification.

On a Geometric Configuration of Two Intersecting Circles: A Partial Result

We study a configuration involving two intersecting circles and prove that the orthocenter of a certain triangle lies on the perpendicular bisector of a segment formed by the intersections of the line of centers with the circles. This is a step towards proving the full statement that a line parallel to a certain line through this orthocenter is tangent to the circumcircle of a triangle formed by the other intersection points.

An Analytic Proof of a Geometry Theorem on Two Intersecting Circles

We prove the theorem using coordinate geometry. Assign coordinates, compute points, show that line through H parallel to AP is tangent to circumcircle of BEF.