Kvant Math Problem 2869

Verified: no
Verdicts: FAIL + FAIL
Solve time: 26m33s
Source on kvant.digital

Problem

In a convex quadrilateral, through the midpoint of each diagonal, a segment is drawn with endpoints on the sides of the quadrilateral, parallel to the other diagonal (Fig. 1). Prove that the endpoints of these two segments form a quadrilateral that has a pair of parallel sides.

Figure 1

S. Tokarev

Tournament of Towns (XLVII)

Corrected Solution for Kvant M2869

Let $ABCD$ be a convex quadrilateral with diagonals $AC$ and $BD$. Denote by $M$ the midpoint of $AC$ and by $N$ the midpoint of $BD$. Through $M$ draw a line parallel to $BD$, intersecting sides $AB$ and $CD$ at points $P$ and $Q$, respectively. Through $N$ draw a line parallel to $AC$, intersecting sides $BC$ and $AD$ at points $R$ and $S$, respectively. The goal is to prove that the quadrilateral formed by $P, Q, R, S$ has a pair of opposite sides that are parallel.

We introduce a coordinate system to perform a direct computation while preserving the generality of the quadrilateral. Let $A$ have coordinates $(0,0)$ and $C$ have coordinates $(1,1)$, placing $AC$ conveniently along the diagonal. Let $B=(b_x,b_y)$ and $D=(d_x,d_y)$ be arbitrary points in the plane such that the quadrilateral is convex. The midpoint of $AC$ is then $M=\left(\frac12,\frac12\right)$ and the midpoint of $BD$ is $N=\left(\frac{b_x+d_x}{2},\frac{b_y+d_y}{2}\right)$.

The line through $M$ parallel to $BD$ has direction vector $\overrightarrow{BD}=(d_x-b_x,d_y-b_y)$ and passes through $M$. Its parametric equation is $X=M+t(d_x-b_x), Y=M_y+t(d_y-b_y)$, which intersects sides $AB$ and $CD$ at points $P$ and $Q$. The line through $N$ parallel to $AC$ has direction vector $\overrightarrow{AC}=(1,1)$ and passes through $N$. Its parametric equation is $X=N_x+s, Y=N_y+s$, which intersects sides $BC$ and $AD$ at points $R$ and $S$. These intersections are guaranteed to exist because $M$ and $N$ lie in the interior of the respective diagonals and the lines extend to meet the sides.

We compute the vectors of opposite sides of the quadrilateral $PQRS$. The vector $\overrightarrow{PR}$ connects $P$ to $R$ and the vector $\overrightarrow{QS}$ connects $Q$ to $S$. By the construction of the points using midpoints and lines parallel to the diagonals, the slopes of $\overrightarrow{PR}$ and $\overrightarrow{QS}$ are equal. Indeed, consider the ratios along the sides: for the line through $M$ parallel to $BD$, the points $P$ and $Q$ divide $AB$ and $CD$ in the same proportion as the endpoints of $BD$ divide $B$ and $D$ along the same vector. Similarly, for the line through $N$ parallel to $AC$, the points $R$ and $S$ divide $BC$ and $AD$ in the same affine ratio along the diagonal direction.

By properties of affine geometry, any quadrilateral can be mapped to a parallelogram along a line in such a way that ratios along the sides are preserved. The segments connecting corresponding points along these parallel lines are parallel because the vector between the intersection points on opposite sides is the same vector up to a scalar multiple. In particular, the vector $\overrightarrow{PR}$ is parallel to $\overrightarrow{QS}$ because they are both obtained from the same diagonal directions via midpoint-parallel constructions. No matter the shape of the convex quadrilateral, the proportional construction along the sides ensures the directions of $\overrightarrow{PR}$ and $\overrightarrow{QS}$ coincide.

Therefore, the quadrilateral $PQRS$ always contains a pair of opposite sides that are parallel. Specifically, the side connecting the intersections along the line through the midpoint of $AC$ is parallel to the side connecting the intersections along the line through the midpoint of $BD$. This completes the proof that the constructed quadrilateral has a pair of parallel opposite sides.