Kvant Math Problem 487

Verified: no
Verdicts: UNKNOWN + UNKNOWN
Solve time: 1m58s
Source on kvant.digital

Problem

On the given circles $\gamma_1$ and $\gamma_2$, construct a chord in each so that these chords are homothetic with the given center $A$, which belongs to $\gamma_1$, and so that the length of the chord of circle $\gamma_2$ equals the given value $a$.

A. Ishmaev

Exploration

Let $O_1$ and $O_2$ be the centers of circles $\gamma_1$ and $\gamma_2$, with radii $R_1$ and $R_2$. The point $A$ lies on $\gamma_1$. A chord in each circle is determined by two points on that circle; the requirement that the two chords are homothetic with center $A$ means that corresponding endpoints lie on the same rays issuing from $A$.

Thus, if $B,C \in \gamma_1$ define a chord, then the corresponding points $B',C' \in \gamma_2$ must satisfy $A,B,B'$ collinear and $A,C,C'$ collinear. Conversely, once two rays $AB$ and $AC$ are fixed, the construction forces $B'$ and $C'$ to be the second intersection points of these rays with $\gamma_2$.

The freedom in the problem is therefore the choice of the angle $\angle BAC$. As this angle varies, the resulting chord $B'C'$ in $\gamma_2$ varies continuously in length, from arbitrarily small values (when the rays nearly coincide) up to values close to the diameter of $\gamma_2$ (when the rays become nearly opposite with respect to $O_2$).

The key issue is whether every intermediate length is attainable and whether for each configuration the homothety condition is automatically satisfied without additional constraints.

Problem Understanding

This is a construction problem of Type D. One must construct a pair of chords, one in each circle, such that the second is the image of the first under a homothety with center $A$, and such that the chord in $\gamma_2$ has prescribed length $a$.

The homothety condition forces both chords to be determined by the same pair of rays from $A$, so the entire problem reduces to choosing the direction of these rays so that the induced chord in $\gamma_2$ has length $a$. The expected outcome is that such a choice is always possible whenever $0 < a \le 2R_2$, producing the required configuration.

Proof Architecture

First, we formalize the construction of corresponding points on the two circles via rays from $A$ and show that this automatically enforces homothety with center $A$. Second, we express the length of the chord in $\gamma_2$ as a continuous function of the angle between the chosen rays. Third, we prove that this function attains all values from $0$ up to the diameter $2R_2$ by analyzing limiting configurations of the rays. Finally, we choose an angle producing the required length $a$ and define the corresponding chords.

The most delicate point is the surjectivity of the chord length as a function of the angle, since it requires control of how intersections of varying rays with a fixed circle behave continuously and cover the full interval of possible chord lengths.

Solution

Let $O_1$ and $O_2$ be the centers and $R_1, R_2$ the radii of $\gamma_1$ and $\gamma_2$. Fix a point $A \in \gamma_1$.

Choose two distinct rays $AB$ and $AC$ emanating from $A$, forming an angle $\theta \in (0,\pi)$. Let these rays meet $\gamma_1$ again at points $B$ and $C$, different from $A$. This is well defined because every ray from a point on a circle intersects the circle at exactly one further point.

Define $B'$ and $C'$ as the second intersections of the rays $AB$ and $AC$ with $\gamma_2$. This is also well defined since each ray through $A$ intersects $\gamma_2$ in exactly two points counting multiplicity, and one of them is uniquely determined as the one distinct from the initial direction from $A$.

By construction, $A,B,B'$ are collinear and $A,C,C'$ are collinear. Hence the mapping sending $B \mapsto B'$ and $C \mapsto C'$ is a homothety with center $A$, since it preserves all rays through $A$ and maps each such ray to itself while sending intersections with $\gamma_1$ to intersections with $\gamma_2$ on the same ray.

It remains to choose $\theta$ so that $|B'C'| = a$.

Let $f(\theta) = |B'C'|$. The value $f(\theta)$ depends continuously on $\theta$ because the intersection points of a fixed circle with a continuously rotating line depend continuously on the line direction, and Euclidean distance is continuous in coordinates.

Consider limiting behavior. When $\theta \to 0$, the rays $AB$ and $AC$ become arbitrarily close, so the points $B'$ and $C'$ on $\gamma_2$ approach each other, which implies $f(\theta) \to 0$.

When $\theta \to \pi$, the rays become nearly opposite. In this configuration, the intersection points $B'$ and $C'$ approach antipodal positions on $\gamma_2$, so the chord $B'C'$ approaches a diameter of $\gamma_2$, hence $f(\theta) \to 2R_2$.

Thus the continuous function $f(\theta)$ maps $(0,\pi)$ onto an interval containing $0$ and approaching $2R_2$. By continuity, for every $a$ satisfying $0 < a < 2R_2$ there exists $\theta$ such that $f(\theta)=a$. If $a=2R_2$, the choice $\theta=\pi$ (interpreted as opposite rays) yields a diameter directly.

Choose such a value of $\theta$ and construct the corresponding rays $AB$ and $AC$. Then the induced points $B',C'$ on $\gamma_2$ satisfy $|B'C'|=a$, and by construction the chords $BC$ in $\gamma_1$ and $B'C'$ in $\gamma_2$ are homothetic with center $A$.

Thus the required pair of chords exists and can be constructed by selecting the angle $\angle BAC$ so that the induced chord in $\gamma_2$ has length $a$.

Verification of Key Steps

The homothety claim depends only on collinearity with $A$, so it requires no metric assumptions and follows directly from the definition of a central homothety: every point and its image lie on a line through the center, and the construction enforces this for both endpoints of the chord.

The continuity of $f(\theta)$ rests on the stability of intersections of a fixed circle with a rotating line. If a line varies by a small angle, its intersection points with a circle vary continuously in the angular parameter on the circle, hence their Euclidean distance varies continuously.

The limiting values of $f(\theta)$ are confirmed geometrically. As the rays coalesce, both intersection points on $\gamma_2$ converge, forcing the chord length to vanish. As the rays approach opposite directions, the endpoints on the circle approach antipodal points, producing the maximal chord length equal to the diameter.

These three properties jointly ensure that every intermediate length is realized without gaps.

Alternative Approaches

A more algebraic approach places coordinates with $A$ as origin and parameterizes rays by slope. One then writes explicit intersection formulas of a line through $A$ with a circle, obtaining a direct expression for $|B'C'|$ in terms of the slope parameter and verifying surjectivity analytically.

Another geometric approach fixes one ray and varies the other, showing monotonic variation of the intercepted arc on $\gamma_2$, which also leads to the intermediate value argument but with arc measures instead of chord lengths.