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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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

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.

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.