From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$. a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$. b) Find the locus of $P$ as we move the 3 arbitary lines.
Problem
Source: Iran 2005
Tags: geometry, rectangle, geometry proposed
darij grinberg
27.08.2005 16:56
As for the source of the problem: [url=http://groups.yahoo.com/group/Hyacinthos/message/70?expand=1]Hyacinthos message #70[/url] wrote: From: Antreas P. Hatzipolakis Date: Tue Jan 4, 2000 7:39pm Subject: 9PC as locus (was: Griffiths Locus) Bernard Gibert wrote: >2). this particular one is described and noted (Ko) in "Le cercle d'Euler" >by Collet & Griso. Dear Bernard I have a question about a locus which probably is included in the book above, a book which I heve not seen: Who is the author of this problem ? Through the vertices of a triangle ABC we draw three parallels, and three other paerpendicular to previous ones. So there are formed three rectangles ADBE, BSCZ, and CKAH, which have as their diagonals the sides AB, BC, CA, of the triangle. Find the locus of the intersection point of the three other diagonals ED, SZ, KH. (word-by-word translation from Panakis (*)) E A K B D S Z H C The locus is the Euler Circle (as we [=Greeks, following you French (**)], call the nine-point-circle or Feuerbach circle). The triangle ABC is inscribed in the rectangle EKCZ. It would be interesting the particular case that the rectangle is a square. Asterisks: (*) Panakis = I. Panakis: 2500 Problems of Geometric Loci With Their Solutions [in Greek]. Athens, ca 1965, p. 654, #582. The book has a general bibliography but not references for each one problem. (**): Cf Victor Thebault: A triangle ABC is inscribed in a circle (O) with a fixed diameter D, and a transversal D', which turns about a fixed point, cuts BC, CA, AB in A1, B1, C1. Let A2 and A3, B2 and B3, C2 and C3 be the orthogonal projections of A and A1, B and B1, C and C1 on D. (a) Show that the circles with centers at the the midpoints of AA1, BB1, CC1 and passing through A2 and A3, and B2 and and B3, C2 and C3 meet in a fixed point on the Euler circle, that is, the nine-point circle of ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ the triangle. (b) Find the locus of the second point of interesection of the three circles. (The American Mathematical Monthly 45(1938) 554 by V. Thebault) Antreas
Amir.S
27.08.2005 17:15
easy problem but nice I just give a hint: Use ceva sinus theorem in median triangle of $ABC$. let the intersection point of diagonals be $P$ with angle chasing show vertix of median triangle and $P$ are on a circle
jred
21.08.2016 17:06
Is there a solution that will handle so many different configurations?