Difference between revisions of "2011 USAJMO Problems/Problem 5"

Latest revision as of 16:00, 14 April 2024

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 $\overline{AC}$ .

The Power of Angle Chasing

https://www.youtube.com/watch?v=Dn1IIx9Cnqw&list=PLqgsN351HEtHgi3Ax2oXJk_bEiROCvNF5

Solution

Since $\overline{DE} \parallel \overline{AC}$ , we have that $AD=CE\rightarrow \angle ABD=\angle CBE$ . But it is well known that $ABCD$ is a harmonic quadrilateral, thus $\overline{BD}$ is a symmedian of triangle $ABC$ , from which it follows that $\overline{BE}$ is a median of $\triangle ABC$ . $\blacksquare$

Solution 1

Connect segment PO, and name the interaction of PO and the circle as point M.

Since PB and PD are tangent to the circle, it's easy to see that M is the midpoint of arc BD.

∠ BOA = 1/2 arc AB + 1/2 arc CE

Since AC // DE, arc AD = arc CE,

thus, ∠ BOA = 1/2 arc AB + 1/2 arc AD = 1/2 arc BD = arc BM = ∠ BOM

Therefore, PBOM is cyclic, ∠ PFO = ∠ OBP = 90°, AF = AC (F is the interaction of BE and AC)

BE bisects AC, proof completed!

~ MVP Harry

Solution 2

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 3

This is the solution from EGMO Problem 1.43 page 242

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 Euclid's Parallel Postulate).

-Evan Chen (vEnhance)

Solution 4

Note that by Lemma 9.9 of EGMO, $(A,C;B,D)$ is a harmonic bundle. We project through $E$ onto $\overline{AC}$ , $-1=(A,C;B,D)\stackrel{E}{=}(A,C;M,P_{\infty})$ Where $P_{\infty}$ is the point at infinity for parallel lines $\overline{DE}$ and $\overline{AC}$ . Thus, we get $\frac{MA}{MC}=-1$ , and $M$ is the midpoint of $AC$ . ~novus677

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 3: / Line 3: @@
 Points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math> lie on a circle <math>\omega</math> and point <math>P</math> lies outside the circle.  The given points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> are collinear, and (iii) <math>\overline{DE} \parallel \overline{AC}</math>.  Prove that <math>\overline{BE}</math> bisects <math>\overline{AC}</math>.
-== Solution ==
+== The Power of Angle Chasing ==
+https://www.youtube.com/watch?v=Dn1IIx9Cnqw&list=PLqgsN351HEtHgi3Ax2oXJk_bEiROCvNF5
-==Solution 1==
+==Solution==
+Since <math>\overline{DE} \parallel \overline{AC}</math>, we have that <math>AD=CE\rightarrow \angle ABD=\angle CBE</math>. But it is well known that <math>ABCD</math> is a harmonic quadrilateral, thus <math>\overline{BD}</math> is a symmedian of triangle <math>ABC</math>, from which it follows that <math>\overline{BE}</math> is a median of <math>\triangle ABC</math>. <math>\blacksquare</math>
+== Solution 1 ==
+Connect segment PO, and name the interaction of PO and the circle as point M.
+Since PB and PD are tangent to the circle, it's easy to see that M is the midpoint of arc BD.
+∠ BOA = 1/2 arc AB + 1/2 arc CE
+Since AC // DE, arc AD = arc CE,
+thus, ∠ BOA = 1/2 arc AB + 1/2 arc AD = 1/2 arc BD = arc BM = ∠ BOM
+Therefore, PBOM is cyclic, ∠ PFO = ∠ OBP = 90°, AF = AC (F is the interaction of BE and AC)
+BE bisects AC, proof completed!
+~ MVP Harry
+==Solution 2==
 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 OPB = \angle OPD = y </math>, <math> \angle BED = \frac{\angle DOB}{2} = 90-y </math>,
-<math>\angle BED = frac{\angle DOB}{2} = 90-y</math>
+<math> \angle ODE = \angle PDE - 90 = 90-x-y </math>
-<math>\angle ODE = \angle PDE - 90 = 90-x-y</math>
+<math> \angle OBE = \angle PBE - 90 = x = \angle OPA </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>.
@@ Line 18: / Line 38: @@
 ~pandadude
-==Solution 2==
+==Solution 3==
+This is the solution from EGMO Problem 1.43 page 242
 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>.
@@ Line 64: / Line 85: @@
 </asy>
-Since quadrilateral <math>BOMP</math> is cyclic, <math>\angle BMP = \angle BOP</math>.  Triangles <math>BOP</math> and <math>DOP</math> are congruent, so <math>\angle BOP = \angle BOD/2 = \angle BED</math>, so <math>\angle BMP = \angle BED</math>. Because <math>AC</math> and <math>DE</math> are parallel, <math>M</math> lies on <math>BE</math> (using Euler's Parallel Postulate).
+Since quadrilateral <math>BOMP</math> is cyclic, <math>\angle BMP = \angle BOP</math>.  Triangles <math>BOP</math> and <math>DOP</math> are congruent, so <math>\angle BOP = \angle BOD/2 = \angle BED</math>, so <math>\angle BMP = \angle BED</math>. Because <math>AC</math> and <math>DE</math> are parallel, <math>M</math> lies on <math>BE</math> (using Euclid's Parallel Postulate).
+-Evan Chen (vEnhance)
+==Solution 4==
+Note that by Lemma 9.9 of EGMO, <math>(A,C;B,D)</math> is a harmonic bundle. We project through <math>E</math> onto <math>\overline{AC}</math>,
+<cmath>-1=(A,C;B,D)\stackrel{E}{=}(A,C;M,P_{\infty})</cmath>
+Where <math>P_{\infty}</math> is the point at infinity for parallel lines <math>\overline{DE}</math> and <math>\overline{AC}</math>. Thus, we get <math>\frac{MA}{MC}=-1</math>, and <math>M</math> is the midpoint of <math>AC</math>. ~novus677
 {{MAA Notice}}

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