2008 USAMO Problems/Problem 2

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Problem

(Zuming Feng) Let $ABC$ be an acute, scalene triangle, and let $M$ , $N$ , and $P$ be the midpoints of $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ , respectively. Let the perpendicular bisectors of $\overline{AB}$ and $\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$ , inside of triangle $ABC$ . Prove that points $A$ , $N$ , $F$ , and $P$ all lie on one circle.

1 Problem
2 Solution
3 Resources

Solution

Solution 1 (synthetic)

Without loss of generality $AB < AC$ . The intersection of $NE$ and $PD$ is $O$ , the circumcenter of $\triangle ABC$ .

Let $\angle BAM = y$ and $\angle CAM = z$ . Note $D$ lies on the perpendicular bisector of $AB$ , so $AD = BD$ . So $\angle FBC = \angle B - \angle ABD = B - y$ . Similarly, $\angle FCE = C - z$ , so $\angle BFC = 180 - (B + C) + (y + z) = 2A$ . Notice that $\angle BOC$ intercepts the minor arc $BC$ in the circumcircle of $\triangle ABC$ , which is double $\angle A$ . Hence $\angle BFC = \angle BOC$ , so $FOBC$ is cyclic.

Lemma 1: $\triangle FEO$ is directly similar to $\triangle NEM$ $\angle OFE = \angle OFC = \angle OBC = \frac {1}{2}\cdot (180 - 2A) = 90 - A$ since $F$ , $E$ , $C$ are collinear, $FOBC$ is cyclic, and $OB = OC$ . Also $\angle ENM = 90 - \angle MNC = 90 - A$ because $NE\perp AC$ , and $MNP$ is the medial triangle of $\triangle ABC$ so $AB \parallel MN$ . Hence $\angle OFE = \angle ENM$ .

Notice that $\angle AEN = 90 - z = \angle CEN$ since $NE\perp BC$ . $\angle FED = \angle MEC = 2z$ . Then $\angle FEO = \angle FED + \angle AEN = \angle CEM + \angle CEN = \angle NEM$ Hence $\angle FEO = \angle NEM$ .

Hence $\triangle FEO$ is similar to $\triangle NEM$ by AA similarity. It is easy to see that they are oriented such that they are directly similar. End Lemma 1.

By the similarity in Lemma 1, $FE: EO = NE: EM\implies FE: EN = OE: NM$ . $\angle FEN = \angle OEM$ so $\triangle FEN\sim\triangle OEM$ by SAS similarity. Hence $\angle EMO = \angle ENF = \angle ONF$ Using essentially the same angle chasing, we can show that $\triangle PDM$ is directly similar to $\triangle FMO$ . It follows that $\triangle PDF$ is directly similar to $MDO$ . So $\angle EMO = \angle DMO = \angle DPF = \angle OPF$ Hence $\angle OPF = \angle ONF$ , so $FONP$ is cyclic. In other words, $F$ lies on the circumcircle of $\triangle PON$ . Note that $\angle ONA = \angle OPA = 90$ , so $APON$ is cyclic. In other words, $A$ lies on the circumcircle of $\triangle PON$ . $A$ , $P$ , $N$ , $O$ , and $F$ all lie on the circumcircle of $\triangle PON$ . Hence $A$ , $P$ , $F$ , and $N$ lie on a circle, as desired.

Solution 2 (synthetic)

Hint: consider $CF$ intersection with $PM$ ; show that the resulting intersection lies on the desired circle. This solution is incomplete. You can help us out by completing it.

Solution 3 (synthetic)

This solution utilizes the phantom point method. Clearly, APON are cyclic because $\angle OPA = \angle ONA = 90$ . Let the circumcircles of triangles $APN$ and $BOC$ intersect at $F'$ and $O$ .

Lemma. If $A,B,C$ are points on circle $\omega$ with center $O$ , and the tangents to $\omega$ at $B,C$ intersect at $Q$ , then $AP$ is the symmedian from $A$ to $BC$ .

This is fairly easy to prove (as H, O are isogonal conjugates, plus using SAS similarity), but the author lacks time to write it up fully, and will do so soon.

It is easy to see $Q$ (the intersection of ray $OM$ and the circumcircle of $\triangle BOC$ ) is colinear with $A$ and $F'$ , and because line $OM$ is the diameter of that circle, $\angle QBO = \angle QCO = 90$ , so $Q$ is the point $Q$ in the lemma; hence, we may apply the lemma. From here, it is simple angle-chasing to show that $F'$ satisfies the original construction for $F$ , showing $F=F'$ ; we are done. This solution is incomplete. You can help us out by completing it.

Solution 4 (trigonometric)

By the Law of Sines, $\frac {\sin\angle BAM}{\sin\angle CAM} = \frac {\sin B}{\sin C} = \frac bc = \frac {b/AF}{c/AF} = \frac {\sin\angle AFC\cdot\sin\angle ABF}{\sin\angle ACF\cdot\sin\angle AFB}$ . Since $\angle ABF = \angle ABD = \angle BAD = \angle BAM$ and similarly $\angle ACF = \angle CAM$ , we cancel to get $\sin\angle AFC = \sin\angle AFB$ . Obviously, $\angle AFB + \angle AFC > 180^\circ$ so $\angle AFC = \angle AFB$ .

Then $\angle FAB + \angle ABF = 180^\circ - \angle AFB = 180^\circ - \angle AFC = \angle FAC + \angle ACF$ and $\angle ABF + \angle ACF = \angle A = \angle FAB + \angle FAC$ . Subtracting these two equations, $\angle FAB - \angle FCA = \angle FCA - \angle FAB$ so $\angle BAF = \angle ACF$ . Therefore, $\triangle ABF\sim\triangle CAF$ (by AA similarity), so a spiral similarity centered at $F$ takes $B$ to $A$ and $A$ to $C$ . Therefore, it takes the midpoint of $\overline{BA}$ to the midpoint of $\overline{AC}$ , or $P$ to $N$ . So $\angle APF = \angle CNF = 180^\circ - \angle ANF$ and $APFN$ is cyclic.

Solution 5 (isogonal conjugates)

Construct $T$ on $AM$ such that $\angle BCT = \angle ACF$ . Then $\angle BCT = \angle CAM$ . Then $\triangle AMC\sim\triangle CMT$ , so $\frac {AM}{CM} = \frac {CM}{TM}$ , or $\frac {AM}{BM} = \frac {BM}{TM}$ . Then $\triangle AMB\sim\triangle BMT$ , so $\angle CBT = \angle BAM = \angle FBA$ . Then we have

$\angle CBT = \angle ABF$ and $\angle BCT = \angle ACF$ . So $T$ and $F$ are isogonally conjugate. Thus $\angle BAF = \angle CAM$ . Then

$\angle AFB = 180 - \angle ABF - \angle BAF = 180 - \angle BAM - \angle CAM = 180 - \angle BAC$ .

If $O$ is the circumcenter of $\triangle ABC$ then $\angle BFC = 2\angle BAC = \angle BOC$ so $BFOC$ is cyclic. Then $\angle BFO = 180 - \angle BOC = 180 - (90 - \angle BAC) = 90 + \angle BAC$ .

Then $\angle AFO = 360 - \angle AFB - \angle BFO = 360 - (180 - \angle BAC) - (90 + \angle BAC) = 90$ . Then $\triangle AFO$ is a right triangle.

Now by the homothety centered at $A$ with ratio $\frac {1}{2}$ , $B$ is taken to $P$ and $C$ is taken to $N$ . Thus $O$ is taken to the circumcenter of $\triangle APN$ and is the midpoint of $AO$ , which is also the circumcenter of $\triangle AFO$ , so $A,P,N,F,O$ all lie on a circle.

Solution 6 (symmedians)

Median $AM$ of a triangle $ABC$ implies $\frac {\sin{BAM}}{\sin{CAM}} = \frac {\sin{B}}{\sin{C}}$ . Trig ceva for $F$ shows that $AF$ is a symmedian. Then $FP$ is a median, use the lemma again to show that $AFP = C$ , and similarly $AFN = B$ , so you're done. This solution is incomplete. You can help us out by completing it.

Solution 7 (inversion)

We consider an inversion by an arbitrary radius about $A$ . We want to show that $P', F',$ and $N'$ are collinear. Notice that $D', A,$ and $P'$ lie on a circle with center $B'$ , and similarly for the other side. We also have that $B', D', F', A$ form a cyclic quadrilateral, and similarly for the other side. By angle chasing, we can prove that $A B' F' C'$ is a parallelogram, indicating that $F'$ is the midpoint of $P'N'$ . This solution is incomplete. You can help us out by completing it.

Solution 8 (analytical)

We let $A$ be at the origin, $B$ be at the point $(a,0)$ , and $C$ be at the point $(b,c):\ b<a$ . Then the equation of the perpendicular bisector of $\overline{AB}$ is $x = a/2$ , and This solution is incomplete. You can help us out by completing it.