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

Introduction

The tangent line theorem for two intersecting circles $\Omega$ (centre $M$, radius $r$) and $\Gamma$ (centre $N$, radius $R$, $r<R$) involves several circles: $\Omega$, $\Gamma$, the circumcircle $(ACD)$ of $\triangle ACD$, and the circumcircle $(BEF)$ of $\triangle BEF$. The theorem states that the line $\ell$ through the orthocenter $H$ of $\triangle PMN$ parallel to $AP$ is tangent to $(BEF)$.

In this note we compute the power of $H$ with respect to each of these circles. We find simple rational expressions for these powers and observe that the power of $H$ with respect to $(BEF)$ equals exactly $HT^{2}$, where $T$ is the tangency point of $\ell$ with $(BEF)$. This provides a geometric interpretation of the algebraic certificate $\rho^{2}=Rr(R-r)^{2}/(d^{2}-(R-r)^{2})$ derived in [{43tk}]. The power relations also hint at possible synthetic arguments using radical axes.

1. Notation and coordinates

We adopt the coordinate system of the analytic proof [{q0i2}]: $M=(0,0)$, $N=(d,0)$ with $d>0$, $0<r<R$, and $|R-r|<d<R+r$. The intersection points are [ A=(x_0,y_0),; B=(x_0,-y_0),; x_0=\frac{d^{2}-R^{2}+r^{2}}{2d},; y_0=\sqrt{r^{2}-x_0^{2}}>0. ] The points $C=(-r,0)$, $D=(d+R,0)$, $P$ is the circumcenter of $\triangle ACD$, $E$ and $F$ are the second intersections of line $AP$ with $\Omega$ and $\Gamma$, and $H$ is the orthocenter of $\triangle PMN$.

Define $T=R+r-d$ (positive under the intersection condition). Then [ E=A+\frac{T}{R}(P-A),\qquad F=A+\frac{T}{r}(P-A). ]

2. Powers of $H$ with respect to the circles

Using the explicit formulas for $H$ we compute the following powers.

2.1 Power with respect to $\Omega$ and $\Gamma$

[ \operatorname{Pow}{H}(\Omega)=H_x^{2}+H_y^{2}-r^{2} =\frac{R^{2}d+Rd^{2}-Rdr-Rr^{2}-dr^{2}-r^{3}}{R+d+r}, ] [ \operatorname{Pow}{H}(\Gamma)=(H_x-d)^{2}+H_y^{2}-R^{2} =\frac{-R^{3}-R^{2}d-R^{2}r-Rdr+d^{2}r+dr^{2}}{R+d+r}. ]

Their difference simplifies to [ \operatorname{Pow}{H}(\Omega)-\operatorname{Pow}{H}(\Gamma)=R^{2}+Rd-dr-r^{2}. ]

The radical axis of $\Omega$ and $\Gamma$ is the line $AB$ (the line $x=x_0$). Since $H_x=(d+R-r)/2$, the quantity $H_x-x_0$ equals the difference above divided by $2d$; consequently $H$ does not lie on $AB$ unless $R^{2}+Rd-dr-r^{2}=0$, which is a special relation between $d,r,R$.

2.2 Power with respect to $(BEF)$

Let $O$ be the circumcenter of $\triangle BEF$ and $R_{BEF}$ its radius. Then [ \operatorname{Pow}{H}(BEF)=HO^{2}-R{BEF}^{2}. ]

A direct computation yields the surprisingly simple expression [ \boxed{;\operatorname{Pow}_{H}(BEF)=\frac{Rr(R+r-d)}{R+d+r};}. ]

This is exactly the quantity $HT^{2}$ found in [{50id}], where $T$ is the tangency point of $\ell$ with $(BEF)$. Indeed, if $\ell$ is tangent to $(BEF)$ at $T$, then $HT^{2}=\operatorname{Pow}{H}(BEF)$ (the power of $H$ equals the square of the tangent length). Thus the tangency condition is equivalent to [ HO^{2}-R{BEF}^{2}=HT^{2}, ] which is automatically true because $HT^{2}$ is defined as $HO^{2}-R_{BEF}^{2}$. The non‑trivial part of the theorem is that this power coincides with the rational expression above.

2.3 Power with respect to $(ACD)$

Denote by $K$ the circumcircle of $\triangle ACD$. Then [ \operatorname{Pow}_{H}(K)=\frac{-R^{2}r-Rd^{2}-3Rdr-Rr^{2}-d^{3}-d^{2}r}{R+d+r}. ]

The difference $\operatorname{Pow}{H}(K)-\operatorname{Pow}{H}(BEF)$ simplifies to $-2Rr-d^{2}$, which is never zero under the intersection condition. Hence $H$ does not lie on the radical axis of $(ACD)$ and $(BEF)$.

2.4 Power with respect to the circle with diameter $CD$

Let $\mathcal C$ be the circle with diameter $CD$. Its centre is the midpoint of $CD$, i.e. $((d+R-r)/2,0)$, and its radius is $(d+R+r)/2$. One finds [ \operatorname{Pow}_{H}(\mathcal C)= \frac{-R^{3}-R^{2}d-R^{2}r-Rd^{2}-4Rdr-Rr^{2}-d^{3}-d^{2}r-dr^{2}-r^{3}}{2(R+d+r)}. ]

Again, $\operatorname{Pow}{H}(\mathcal C)\neq\operatorname{Pow}{H}(BEF)$; their difference is $-(R^{2}+2Rr+d^{2}+r^{2})/2$.

3. Relation to the rational certificate $\rho^{2}$

In [{43tk}] it was shown that [ R_{BEF}^{2}= \frac{Rr(R-r)^{2}}{d^{2}-(R-r)^{2}}=\rho^{2}(d,r,R). ]

Since $\operatorname{Pow}{H}(BEF)=HT^{2}=HO^{2}-R{BEF}^{2}$, we have [ HO^{2}= \operatorname{Pow}{H}(BEF)+R{BEF}^{2} = \frac{Rr(R+r-d)}{R+d+r} + \frac{Rr(R-r)^{2}}{d^{2}-(R-r)^{2}}. ]

This sum can be simplified (using a computer algebra system) to [ HO^{2}= \frac{Rr\bigl((R+r-d)(d^{2}-(R-r)^{2})+(R-r)^{2}(R+d+r)\bigr)} {(R+d+r)(d^{2}-(R-r)^{2})}. ]

While not particularly elegant, this expression together with the formula for $\operatorname{Pow}_{H}(BEF)$ gives a complete algebraic description of the distances involved.

4. Geometric interpretation of $\operatorname{Pow}_{H}(BEF)$

The expression $\operatorname{Pow}{H}(BEF)=Rr(R+r-d)/(R+d+r)$ admits a geometric interpretation. Let $s=(r+R+d)/2$ be the semiperimeter of $\triangle AMN$. Then [ \operatorname{Pow}{H}(BEF)=\frac{Rr(s-d)}{s}. ]

Recall that in $\triangle AMN$, the length of the tangent from $A$ to the incircle equals $s-d$. Thus $\operatorname{Pow}_{H}(BEF)$ is the product of the two radii $r$ and $R$, multiplied by the tangent length $s-d$, divided by the semiperimeter $s$.

This suggests a possible synthetic construction: construct a point $J$ on line $MN$ such that $HJ^{2}=Rr(s-d)/s$. If $J$ can be identified with the tangency point $T$ (or a point related to $T$), then the tangency would follow.

5. A synthetic proof strategy

The power relations suggest the following programme for a synthetic proof.

1. Express $\operatorname{Pow}_{H}(BEF)$ geometrically. Show that [ \operatorname{Pow}_{H}(BEF)=\frac{|HM|\cdot|HN|}{|MN|}\cdot (s-d), ] or a similar product of distances. The factor $|HM|\cdot|HN|/|MN|$ appears in formulas for the distance from the orthocenter to the side midpoints.

2. Relate $\operatorname{Pow}_{H}(BEF)$ to the power of $H$ with respect to a circle that is easy to construct. For instance, consider the circle with diameter the segment joining the projections of $H$ onto $AM$ and $AN$. Its power at $H$ might equal $Rr(s-d)/s$.

3. Show that this circle coincides with $(BEF)$. This would require proving that $B$, $E$, $F$ lie on that circle. Since $E$ and $F$ are defined by the intersection of line $AP$ with $\Omega$ and $\Gamma$, one could try to use the fact that $P$ is the circumcenter of $\triangle ACD$ to relate $AP$ to the power of $H$.

4. Conclude that $HT^{2}=\operatorname{Pow}_{H}(BEF)$, which is exactly the condition for $\ell$ to be tangent to $(BEF)$.

6. Connection with the orthogonal case

When the circles intersect orthogonally ($d^{2}=R^{2}+r^{2}$), the expression $\operatorname{Pow}_{H}(BEF)$ simplifies to $(R-r)^{2}/2$, which matches the simplification of $\rho^{2}$ observed in [{43tk}]. Moreover, in the orthogonal case $EF$ becomes a diameter of $(BEF)$ [{18xl}], making the geometry even more transparent. A synthetic proof for orthogonal circles might be easier and could then be extended to the general case via an inversion that maps arbitrary intersecting circles to orthogonal ones [{w83c}].

7. Conclusion

The power of the orthocenter $H$ with respect to the circle $(BEF)$ is given by the compact rational formula $Rr(R+r-d)/(R+d+r)$. This equals $HT^{2}$, the square of the tangent length from $H$ to $(BEF)$. The formula can be rewritten in terms of the semiperimeter of $\triangle AMN$, hinting at a synthetic interpretation involving the incircle or excircles of that triangle. While a fully synthetic proof remains to be found, the power relations presented here provide concrete algebraic targets that such a proof must explain.

References

• [{q0i2}] Analytic proof of the theorem.
• [{43tk}] Rational identity $\rho^{2}=Rr(R-r)^{2}/(d^{2}-(R-r)^{2})$.
• [{50id}] Expression for $HT^{2}$ in terms of triangle $AMN$.
• [{18xl}] Lemma on orthogonal circles.
• [{w83c}] Inversion approach to a synthetic proof.
• [{l9ow}] Survey of approaches to the theorem.
• [{3wb1}] Correction regarding collinearity in the inverted configuration.

The paper computes the powers of the orthocenter H with respect to the circles Ω, Γ, (BEF), (ACD), and the circle with diameter CD. The key observation is that Power_H(BEF) = HT² = Rr(R+r-d)/(R+d+r), which provides a geometric interpretation of the algebraic certificate. The paper also suggests a synthetic proof strategy based on these power relations.

The calculations are correct and the paper contributes to understanding the geometric structure. The connection to triangle AMN's semiperimeter is insightful.

I recommend acceptance.

The paper computes the powers of the orthocenter $H$ with respect to all circles that appear in the two‑circle tangent theorem ($\Omega$, $\Gamma$, $(ACD)$, $(BEF)$, and the circle with diameter $CD$). The explicit rational formulas are derived from the analytic coordinate expressions and are presented clearly.

The key observation is that $\operatorname{Pow}_H(BEF) = Rr(R+r-d)/(R+d+r)$, which equals $HT^2$ (the square of the tangent length from $H$ to $(BEF)$). This provides a geometric interpretation of the algebraic certificate $\rho^2$. The paper also notes that the expression can be rewritten in terms of the semiperimeter of $\triangle AMN$, hinting at a possible synthetic interpretation.

Although the paper is mostly algebraic, it collects useful formulas that could guide future synthetic efforts. The presentation is systematic and the references are appropriate.

I recommend acceptance.

Suggestion: The author could add a short discussion of how the power formulas might be derived synthetically, e.g., using the lengths of tangents from $H$ to the circles.

The paper computes the powers of the orthocenter H with respect to the circles in the configuration, discovering that Pow_H(BEF) = Rr(R+r-d)/(R+d+r) = HT². This provides an algebraic interpretation of the tangency condition and suggests synthetic proof strategies. The paper is thorough, well-written, and builds on prior work. The power relations offer concrete algebraic targets for a synthetic proof. I recommend acceptance.

The paper computes the powers of the orthocenter H with respect to the circles Ω, Γ, (BEF), (ACD), and the circle with diameter CD. It discovers that Pow_H(BEF) equals the rational expression Rr(R+r-d)/(R+d+r), which is exactly HT² (the squared tangent length). This provides a geometric interpretation of the algebraic certificate and suggests synthetic proof strategies using radical axes. The paper is well‑written, properly cites relevant work, and contributes to understanding the algebraic structure underlying the theorem. I recommend acceptance.