sum-and-product-of-chords-from-point

Sum and Product of Chords From a Fixed Point

Summary: §§405–409 within the two-intersection framework: $CM+CN=a$ has no algebraic solution because $P=Mz/L$ carries the irrationality $z=\sqrt{x^2+y^2}$; relaxing to “more than two intersections” gives infinite-family solutions like concentric-circle pairs. By contrast $CM\cdot CN=a^2$ is satisfied by every circle through $C$, and from each higher order one extracts an infinite family $M/N=(x^2+y^2+a^2)/a^2$.

Sources: chapter17 §§405–409.

Last updated: 2026-05-11.

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Sum $CM+CN$ constant: impossible (§405)

From two-point-equation-from-fixed-point, the chord-distances satisfy $z^2-Pz+Q=0$ with $P=Mz/L$. Demanding $CM+CN=a$ constant means $P=a$, i.e.

$\frac{Mz}{L}=a⟺M\sqrt{x^2+y^2}=aL.$

The left side carries the irrationality $\sqrt{x^2+y^2}$, which the right side cannot cancel for a polynomial $L$. Hence no algebraic curve has the property $CM+CN=a$ across all lines through $C$ (when only two intersections are demanded).

This is a genuine no-solution result, not just a technical obstruction: it reflects that constant chord-sum from an interior point is a property of the ellipse from its foci, and any other point fails to inherit the property.

Relaxing to more intersections (§406)

Drop the two-intersection constraint and allow extra intersections coincident with the two at $M,N$. Then $P=a$ becomes $z^2-az+Q=0$ with $Q=N{z}^2/L$, and squaring to remove the irrationality gives

$a^2{z}^2=(z^2+Q{)}^2,\text{i.e.}{a}^2{L}^2=(x^2+y^2)(L+N{)}^2.$

The simplest case $L=x^2+y^2$, $N=\pm b^2$ (constant) yields

$a^2(x^2+y^2)=(x^2+y^2\pm b^2{)}^2$

— a complex fourth-order curve that factors as two concentric circles with $C$ as common center. Next simplest ($L=\alpha x^2+\beta xy+\gamma y^2$, $N=\pm b^2$) gives a sixth-order solvable for $y$:

$y^2+x^2=\frac{a^2{x}^4}{x^4\pm 2b^2{x}^2+b^4},y=\frac{x\sqrt{a^2{x}^2-x^4-(\pm 2b^{3}x)-b^4}}{x^2\pm b^2}.$

(Euler is explicit at §406 that these “satisfy the condition” only in the loosened sense — additional intersections beyond $M,N$ are permitted.)

Product $CM\cdot CN$ constant: the circle (§407)

The product condition is $Q=a^2$, i.e. $N{z}^2/L=a^2$, i.e.

$\frac{N(x^2+y^2)}{L}=a^2⟺L=\frac{N(x^2+y^2)}{a^2}.$

Since $N$ is non-irrational of degree $n$, the product condition is algebraically clean — no irrationality to remove. The simplest case is $N=1$, giving

$L=\frac{x^2+y^2}{a^2}\cdot \text{(unit)},$

and the general equation $L-M+N=0$ specializes to

$\frac{(x^2+y^2)}{a^2}\cdot 1-M+1=0,$

or with $M/N=(\alpha x+\beta y)/a$ (first-degree),

$x^2+y^2-a(\alpha x+\beta y)+a^2=0.$

This is a circle in rectangular coordinates — and §409 notes that no other conic section satisfies the condition, recovering the Euclid-Elements power-of-a-point theorem.

Higher orders (§408)

§408 generalizes: with $L=N(x^2+y^2+a^2)/a^2$ (in particular $M/N=(x^2+y^2+a^2)/a^2$ a first-degree function), the general equation becomes

$N(x^2+y^2)/a^2-M+N=0⟺M{a}^2=N(x^2+y^2+a^2).$

Each choice of degree gives a new family. §409 writes out the third-order instance:

$(\delta x+ϵy)(x^2+y^2)-a(\alpha x^2+\beta xy+\gamma y^2)+a^2(\delta x+ϵy)=0,$

equivalent to $(\alpha x^2+\beta xy+\gamma y^2)/a(\delta x+ϵy)=(x^2+y^2+a^2)/a^2$. From any order, an analogous curve satisfies the constant-product condition.

The asymmetry between sum and product

Why does the sum fail while the product succeeds? The sum-of-roots $P$ is odd (degree-1 in $z$) so it inherits the irrational factor $z=\sqrt{x^2+y^2}$; the product-of-roots $Q$ is even (degree-2 in $z$) and the irrationality cancels. This parity asymmetry — set up in §400 as “P odd, Q even in $\sin \varphi ,\cos \varphi$” — propagates structurally into the constant-value problems.

Related pages

• chapter-17-on-finding-curves-from-other-properties
• two-point-equation-from-fixed-point — the setup.
• constant-chord-interval-conchoid — continues the squared-quantity theme with $C{M}^2+C{N}^2$ and $(CN-CM{)}^2$ constant.
• ordinate-powers-from-fixed-point — extends to $C{M}^n+C{N}^n$ constant.
• chord-rectangle-property — chapter 5 conic analogue.
• two-ordinate-sum-and-product — chapter 16 ordinate analogue.