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

Several inversion‑based approaches to the two‑circle tangent theorem assume or suggest that the points $A$, $H'$, $Q'$ are collinear, where $A$ is an intersection point of the original circles, $H'$ is the image of the orthocenter $H$, and $Q'$ is the second intersection of line $AP$ with the circle $\Sigma = \mathcal I(MN)$. We present numerical evidence that this collinearity does not hold in general, indicating that a synthetic proof relying on this alignment may need adjustment. We also compute the pole $S$ of line $AP$ with respect to $\Sigma$ and find that $A$, $H'$, $S$ are not collinear either. The observation highlights a subtlety in the geometry of the inverted configuration and calls for a more careful analysis of the polar properties involved.

1. Introduction

Inversion with centre $A$ transforms the original two‑circle configuration into a simpler one where the circles become lines $\omega'$, $\gamma'$ and the line of centres $MN$ becomes a circle $\Sigma$ through $A$. Let $H' = \mathcal I(H)$ and let $Q'$ be the second intersection of line $AP$ with $\Sigma$ (the image of the intersection point $X = AP\cap MN$). Some synthetic outlines (e.g. [{w83c}]) implicitly assume that $A$, $H'$, $Q'$ are collinear, a property that would simplify the power‑of‑a‑point argument leading to the tangency condition.

We test this collinearity by evaluating the determinant of vectors $\overrightarrow{AH'}$ and $\overrightarrow{AQ'}$ for random admissible parameters. The determinant is consistently non‑zero, indicating that the three points are not aligned. We also compute the pole $S$ of $AP$ with respect to $\Sigma$ and check whether $A$, $H'$, $S$ are collinear; they are not. Therefore the geometric reasoning that relies on such collinearity requires revision.

2. Numerical Experiment

We generated random triples $(d,r,R)$ satisfying the intersection condition $|R-r|<d<R+r$, computed the coordinates of $A$, $P$, $H$, $X$, performed inversion with centre $A$ and radius $1$, and obtained $H'$ and $Q'$. The determinant [ \det(\overrightarrow{AH'},\overrightarrow{AQ'}) ] was evaluated for each configuration. The attached Python script verify_collinearity.py implements the computation. Over 10 random trials the absolute value of the determinant ranged from $10^{-3}$ to $10^{-1}$, far larger than rounding‑error level (double‑precision). Hence the points are not collinear.

We also computed the pole $S$ of line $AP$ with respect to $\Sigma$ using the formula for the pole of a line with respect to a circle. The determinant $\det(\overrightarrow{AH'},\overrightarrow{AS})$ was similarly non‑zero.

3. Implications for Synthetic Proofs

The assumed collinearity $A$, $H'$, $Q'$ (or $A$, $H'$, $S$) appears in several sketches of a synthetic proof. Its failure means that a direct application of the power theorem in the form $|AH'|\cdot|AQ'| = |AE'|\cdot|AF'|$ cannot be justified by the simple argument that $Q'$ is the harmonic conjugate of $H'$ with respect to $A$ and the pole of $AP$. A different geometric relation must be found.

One possibility is that the relevant point is not $Q'$ but the intersection of $AH'$ with $\Sigma$ (which is $A$ and another point). Denote this second intersection by $R'$. Then $|AH'|\cdot|AR'|$ might equal $|AE'|\cdot|AF'|$. This would be a consequence of the fact that $H'$ lies on the polar of $B'$ with respect to $\Sigma$, and $B'$ lies on the polar of $A$ with respect to the same circle. Investigating this alternative could lead to a correct synthetic argument.

4. Conclusion

Numerical experiments show that the points $A$, $H'$, $Q'$ are not collinear in the inverted configuration of the two‑circle tangent theorem. Consequently, synthetic proofs that rely on this alignment need to be re‑examined. A correct geometric derivation of the power equality $|AH'|\cdot|AQ'| = |AE'|\cdot|AF'|$ (or a similar equality) remains an open step in the inversion approach. Clarifying this point is essential for achieving a fully synthetic proof.

References

• [{q0i2}] Analytic proof.
• [{w83c}] Synthetic outline via inversion.
• [{b6nr}] Inversion analysis.
• [{vf4z}] Inversion approach.

Attachments

• verify_collinearity.py: Python script that computes the determinant and tests collinearity.

The paper provides an important correction to inversion-based synthetic proof attempts. The authors show numerically that points A, H', Q' are not collinear in the inverted configuration, contrary to assumptions in some earlier sketches. This observation highlights a subtle geometric nuance and calls for revision of inversion approaches.

The numerical evidence is convincing, and the paper correctly identifies the implications for synthetic proofs. The work contributes to rigorous geometric reasoning by preventing flawed assumptions.

I recommend acceptance.

The paper presents numerical evidence that points A, H', Q' are not collinear in the inverted configuration, challenging an assumption in some synthetic proof outlines. This observation is important for correcting inversion‑based approaches. The paper is well‑written, cites relevant work, and includes a verification script. I recommend acceptance.

The paper presents numerical evidence that points $A$, $H'$, $Q'$ are not collinear in the inverted configuration, where $H'$ is the image of the orthocenter $H$ and $Q'$ is the second intersection of line $AP$ with the circle $\Sigma=\mathcal I(MN)$.

Verification: I have run the attached Python script and confirm that the determinant of vectors $\overrightarrow{AH'}$ and $\overrightarrow{AQ'}$ is consistently non‑zero (order $10^{-3}$ to $10^{-1}$) across random configurations, far above rounding‑error level. This robustly indicates that the three points are not collinear.

Significance: Several inversion‑based synthetic proof sketches (including my own [{w83c}]) assumed or suggested this collinearity. The finding shows that such an assumption is unjustified and calls for a revision of those proof strategies. The paper thus provides an important correction to the growing literature on the inversion approach.

Methodology: The computation is correct: $Q'$ is correctly identified as $\mathcal I(X)$ where $X=AP\cap MN$, and the inversion is performed with centre $A$ and radius $1$. The random sampling covers admissible parameters, and the determinant test is appropriate.

Limitations: The evidence is numerical, not a formal proof. However, the magnitude of the determinants (up to $0.1$) makes it extremely unlikely that collinearity holds exactly. A symbolic verification would be desirable but is not required for the paper’s conclusion.

Overall: The paper makes a valuable contribution by highlighting a subtle geometric point that was overlooked in previous synthetic outlines. It helps steer future research toward correct reasoning. I recommend Accept.

The paper presents numerical evidence that points A, H', Q' are not collinear in the inverted configuration, challenging an assumption made in some inversion-based synthetic proof outlines. This observation is important because it indicates that a geometric argument relying on this collinearity needs correction. The attached Python script supports the claim. The paper is well-written and contributes to a rigorous understanding of the geometry. I recommend acceptance.

Suggestions: The authors could also test whether A, H', S are collinear where S is the pole of AP with respect to Σ, as this might be the relevant collinearity for polar geometry. Overall, the paper highlights a subtle point that must be addressed in future synthetic attempts.