https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Kathirm&feedformat=atom AoPS Wiki - User contributions [en] 2021-10-18T13:47:31Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=1959_IMO_Problems/Problem_6&diff=98080 1959 IMO Problems/Problem 6 2018-10-08T21:36:23Z <p>Kathirm: </p> <hr /> <div>== Problem ==<br /> <br /> Two [[plane]]s, &lt;math&gt;P &lt;/math&gt; and &lt;math&gt;Q &lt;/math&gt;, [[intersect]] along the [[line]] &lt;math&gt;p &lt;/math&gt;. The point &lt;math&gt;A &lt;/math&gt; is in the plane &lt;math&gt;P &lt;/math&gt;, and the point &lt;math&gt;{C} &lt;/math&gt; is in the plane &lt;math&gt;Q &lt;/math&gt;; neither of these points lies on the straight line &lt;math&gt;p &lt;/math&gt;. Construct an [[isosceles trapezoid]] &lt;math&gt;ABCD &lt;/math&gt; (with &lt;math&gt;AB &lt;/math&gt; [[parallel]] to &lt;math&gt;DC &lt;/math&gt;) in which a [[circle]] can be constructed, and with [[vertex | vertices]] &lt;math&gt;B &lt;/math&gt; and &lt;math&gt;D &lt;/math&gt; lying in the planes &lt;math&gt;P &lt;/math&gt; and &lt;math&gt;Q &lt;/math&gt;, respectively.<br /> <br /> == Solution ==<br /> <br /> We first observe that we must have both lines &lt;math&gt;AB &lt;/math&gt; (which we shall denote &lt;math&gt;a &lt;/math&gt;) and &lt;math&gt;DC &lt;/math&gt; (which we shall denote &lt;math&gt;c &lt;/math&gt;) parallel to &lt;math&gt;p &lt;/math&gt;, since if one of them is not, then neither can be and they must both intersect &lt;math&gt;p &lt;/math&gt; (since they are both coplanar with &lt;math&gt;p &lt;/math&gt;), making them [[skew lines | skew]].<br /> <br /> Now we note since a circle can be inscribed in the trapezoid, we must have &lt;math&gt;AB + DC = AD + BC&lt;/math&gt;, and since the trapezoid is isosceles, this implies that each of the trapezoid's legs has length equal to the [[arithmetic mean|average]] of the lengths of the bases.<br /> <br /> We can find this average by dropping [[perpendicular]] &lt;math&gt;AA' &lt;/math&gt; to &lt;math&gt;c &lt;/math&gt; such that &lt;math&gt;A' &lt;/math&gt; is on &lt;math&gt;c &lt;/math&gt;. The average will be &lt;math&gt;A'C &lt;/math&gt;, which is one of the sides of the [[rectangle]] with sides on &lt;math&gt;a &lt;/math&gt; and &lt;math&gt;c &lt;/math&gt; with vertices at &lt;math&gt;A &lt;/math&gt; and &lt;math&gt;{C} &lt;/math&gt;.<br /> <br /> We now draw a circle with center &lt;math&gt;{C} &lt;/math&gt; that contains &lt;math&gt;A' &lt;/math&gt;. The intersections of this circle with &lt;math&gt;a &lt;/math&gt; are the two possible values of &lt;math&gt;B &lt;/math&gt;, from either of which it is trivial to determine the corresponding location for &lt;math&gt;D &lt;/math&gt;. It is worth noting that the intersection points may concur (in which case there is only one distinct possibility (a square)), or they may not occur at all. Q.E.D.<br /> <br /> <br /> {{alternate solutions}}<br /> <br /> <br /> == See Also ==<br /> <br /> <br /> {{IMO box|year=1959|num-b=5|after=Last question}}<br /> <br /> [[Category: Olympiad Geometry Problems]]<br /> [[Category: 3D Geometry Problems]]<br /> [[Category: Geometric Construction Problems]]</div> Kathirm