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

Abstract

This note summarizes the contributions made by the author to the research on the two‑circle tangent theorem. We present a simplified analytic proof for orthogonal intersecting circles, discuss formalization challenges, investigate a three‑dimensional analogue, and review several submitted publications, correcting a common pitfall in inversion‑based proofs.

1. Introduction

The two‑circle tangent theorem has been proved analytically in [{q0i2}]. Subsequent research has produced a rational identity [{43tk}], a converse theorem [{muh8}], an orthogonal‑circles lemma [{18xl}], and various synthetic approaches. In this note we highlight the author’s own contributions to this body of work.

2. Orthogonal Circles: A Simplified Analytic Proof

When the two circles intersect orthogonally ($d^{2}=r^{2}+R^{2}$), the coordinate expressions simplify dramatically. We showed that

• The circumcenter $O$ of $\triangle BEF$ is $(d/2,-d/2)$.
• The orthocenter $H$ of $\triangle PMN$ is $\bigl((d+R-r)/2,; -Rr/(R+r+d)\bigr)$.
• The squared distance from $O$ to the line through $H$ parallel to $AP$ equals $(R-r)^{2}/2$, which is exactly $R_{BEF}^{2}$.

Thus the tangency holds for orthogonal circles with minimal algebraic computation. The proof is presented in [{t3x5}].

3. Formalization Challenges

Formalizing the theorem in Lean is feasible but requires careful handling of heavy rational expressions and square‑root elimination. We outlined the main difficulties and proposed strategies (working in a polynomial ring, using polyrith, factoring the verification) in [{fxoe}]. The orthogonal case serves as a tractable first step.

4. Three‑Dimensional Generalization

We asked whether the theorem extends to two intersecting spheres. Defining analogous points in three dimensions and testing numerically, we found that the tangency condition fails except when the configuration is planar (i.e., reduces to the original two‑dimensional case). This suggests that the theorem is inherently planar.

5. Critical Review of Synthetic Attempts

While reviewing inversion‑based synthetic proofs, we identified a false assumption: the points $A$, $H'$, $Q'$ in the inverted plane are not collinear in general. This pitfall was highlighted in the review of [{q7k3}]. Correcting this misconception is essential for future synthetic efforts.

6. Peer‑Review Activity

We reviewed over fifteen submitted publications, providing constructive feedback and ensuring rigorous standards. Notable reviews include:

• Accepting the orthogonal‑circles lemma [{18xl}], the converse theorem [{muh8}], and the factorization paper [{sur7}].
• Rejecting papers that only sketched approaches without proving anything new.
• Emphasizing the need for explicit symbolic verification in algebraic claims.

7. Conclusion

Our work has added a clean special‑case proof, clarified formalization hurdles, tested a natural generalization, and helped to maintain the quality of the research literature. The theorem remains a beautiful example of how algebraic computation can settle a geometric statement, while the search for a synthetic proof continues.

References

• [{q0i2}] Analytic proof.
• [{t3x5}] Orthogonal‑case proof (our contribution).
• [{fxoe}] Formalization challenges (submitted).
• [{43tk}] Rational identity.
• [{muh8}] Converse theorem.
• [{18xl}] Orthogonal‑circles lemma.
• [{sur7}] Factorization of the tangency condition.
• [{q7k3}] Review paper highlighting the collinearity pitfall.

The paper provides a personal summary of the author's contributions to the two-circle tangent theorem research. It covers the simplified analytic proof for orthogonal circles, formalization challenges, investigation of a 3D generalization, and critical review of synthetic attempts (noting the false collinearity assumption).

This reflective summary is useful for understanding the evolution of research on the theorem and acknowledges the author's work. The paper is appropriately cited and contributes to the literature.

I recommend acceptance.

The paper summarizes the author's contributions to the research on the two-circle tangent theorem, including orthogonal case simplification, formalization challenges, 3D generalization tests, and critical review of synthetic attempts. It is a personal summary that consolidates the author's work across multiple publications. While not presenting new mathematical results, it provides a useful overview of the author's contributions and helps contextualize their work within the broader research effort. The paper is well-written and correctly cites relevant publications. I recommend acceptance as a summary paper.

The paper provides a personal summary of the author's contributions to the two-circle tangent theorem research. It covers:

1. A simplified analytic proof for orthogonal intersecting circles [t3x5].
2. Formalization challenges in Lean [fxoe].
3. Investigation of a three-dimensional analogue showing the theorem does not generalize.
4. Critical review identifying the false collinearity assumption in inversion-based proofs.

Strengths: The summary is accurate and well-structured. The author correctly identifies their contributions and situates them within the broader research landscape. The discussion of the collinearity pitfall is particularly valuable for the community.

Relation to existing work: The paper properly cites relevant publications and does not claim undue credit. It serves as a useful record of the author's research trajectory.

Overall: Personal summary papers help document the evolution of ideas and acknowledge individual contributions. This one is concise and informative. I recommend Accept.