A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.
Problem
Source: Kazakhstan NMO 2015 (second round) P3
Tags: geometry, rectangle, geometric transformation, geometry proposed
19.01.2015 22:31
The proposition is missing the three sides of the triangle, as it is possible to inscribe the rectangle in such a way that two opposite vertices lie on a side, otherwise the center lies on the line connecting the midpoint of a side with the midpoint of the corresponding altitude. These are the 3 Schwatt lines of ABC concurring at its symmedian point. See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=598678.
20.01.2015 23:49
i think, if we mean by the side the strict meaning which is the segment delimited by two vertices the solution is just the three segments concurrent at the symmedian point ;but if we mean by it the line ,it s ok that the solution include certainly the sides
21.01.2015 15:24
Consider the set of rectangles $TUVW$ with $VW \in BC$ and $T \in AB, U \in AC$. We must have $TU \parallel BC \equiv VW$, and that if $V$ is the projection of $T$ on $BC$, $V \mapsto T \mapsto U$ preserves ratio. Hence, there is a spiral similarity mapping $V \mapsto U$ from the set $BC \mapsto AC$. Hence, the midpoint of $VU$ moves along a line i.e. the centres of the rectangles move along a line. When $T=A=U$, the centre is the midpoint of $AD$, where $D$ is the foot of altitude from $A$ to $BC$. When $T=B, U=C$ then the centre is the midpoint of $BC$. So the locus is the A-Schwatt line. Similarly, we conclude they move on the Schwatt lines. Let the tangents at $B, C$ to $\odot ABC$ meet at $P$, point at infintiy perpendicular to $BC$ be $\ell_A$ and symmedian point of $ABC$ be $L$. If $Q, R$ are midpoints of $BC$ and $AD$, then \[Q(P, L; D, A) \equiv Q(\ell_A, L; D, A) = -1 = Q(\ell_A, R; D, A)\] $\implies L \in QR$. Hence, the concurrence point of the locus lines is the symmedian point.
14.07.2021 21:52
schwatt lines xd. They concur at the Lemoine point.