2011 USAJMO Problems/Problem 5: Difference between revisions

Revision as of 18:37, 4 March 2018

Problem

Points $A$ , $B$ , $C$ , $D$ , $E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$ , (ii) $P$ , $A$ , $C$ are collinear, and (iii) $\overline{DE} \parallel \overline{AC}$ . Prove that $\overline{BE}$ bisects $\overline{AC}$ .

Solution

Solution 1

Let $O$ be the center of the circle, and let $X$ be the intersection of $AC$ and $BE$ . Let $\angle OPA$ be $x$ and $\angle OPD$ be $y$ .

$\angle OPB = \angle OPD = y$ $\angle BED = frac{\angle DOB}{2} = 90-y$ $\angle ODE = \angle PDE - 90 = 90-x-y$ $\angle OBE = \angle PBE - 90 = x = \angle OPA$

Thus $PBXO$ is a cyclic quadrilateral and $\angle OXP = \angle OBP = 90$ and so $X$ is the midpoint of chord $AC$ .

~pandadude

Solution 2

Let $O$ be the center of the circle, and let $M$ be the midpoint of $AC$ . Let $\theta$ denote the circle with diameter $OP$ . Since $\angle OBP = \angle OMP = \angle ODP = 90^\circ$ , $B$ , $D$ , and $M$ all lie on $\theta$ .

$[asy] import graph; unitsize(2 cm); pair A, B, C, D, E, M, O, P; path circ; O = (0,0); circ = Circle(O,1); B = dir(100); D = dir(240); P = extension(B, B + rotate(90)*(B), D, D + rotate(90)*(D)); C = dir(-40); A = intersectionpoint((P--(P + 0.9*(C - P))),circ); E = intersectionpoint((D + 0.1*(C - A))--(D + C - A),circ); M = (A + C)/2; draw(circ); draw(P--B); draw(P--D); draw(P--C); draw(B--E); draw(D--E); draw(O--B); draw(O--D); draw(O--M); draw(O--P); draw(Circle((O + P)/2, abs(O - P)/2),dashed); draw(D--M); dot("$A$", A, NE); dot("$B$", B, NE); dot("$C$", C, SE); dot("$D$", D, S); dot("$E$", E, S); dot("$M$", M, NE); dot("$O$", O, dir(0)); dot("$P$", P, W); label("$\theta$", (O + P)/2 + abs(O - P)/2*dir(120), NW); [/asy]$

Since quadrilateral $BOMP$ is cyclic, $\angle BMP = \angle BOP$ . Triangles $BOP$ and $DOP$ are congruent, so $\angle BOP = \angle BOD/2 = \angle BED$ , so $\angle BMP = \angle BED$ . Because $AC$ and $DE$ are parallel, $M$ lies on $BE$ (using Euler's Parallel Postulate). These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

@@ Line 4: / Line 4: @@
 == Solution ==
+==Solution 1==
+Let <math>O</math> be the center of the circle, and let <math>X</math> be the intersection of <math>AC</math> and <math>BE</math>. Let <math>\angle OPA</math> be <math>x</math> and <math>\angle OPD</math> be <math>y</math>.
+<math>\angle OPB = \angle OPD = y</math>
+<math>\angle BED = frac{\angle DOB}{2} = 90-y</math>
+<math>\angle ODE = \angle PDE - 90 = 90-x-y</math>
+<math>\angle OBE = \angle PBE - 90 = x = \angle OPA</math>
+Thus <math>PBXO</math> is a cyclic quadrilateral and <math>\angle OXP = \angle OBP = 90</math> and so <math>X</math> is the midpoint of chord <math>AC</math>.
+~pandadude
+==Solution 2==
 Let <math>O</math> be the center of the circle, and let <math>M</math> be the midpoint of <math>AC</math>.  Let <math>\theta</math> denote the circle with diameter <math>OP</math>.  Since <math>\angle OBP = \angle OMP = \angle ODP = 90^\circ</math>, <math>B</math>, <math>D</math>, and <math>M</math> all lie on <math>\theta</math>.

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