2016 USAMO Problems/Problem 3

Problem

Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$ -excenter, $C$ -excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY = \angle CBY$ and $\overline{BE}\perp\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ = \angle BCZ$ and $\overline{CF}\perp\overline{AB}.$

Lines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.

Solution

This problem can be proved in the following two steps.

1. Let $I_A$ be the $A$ -excenter, then $I_A,O,$ and $P$ are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for $\triangle I_AI_BI_C.$

2. Show that $I_AY^2-I_AZ^2=OY^2-OZ^2,$ which implies $\overline{OI_A}\perp\overline{YZ}.$ This can be proved by multiple applications of the Pythagorean Thm.

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

Solution 2

We find point $T$ on line $YZ,$ we prove that $TY \perp OI_A$ and state that $P$ is the point $X(24)$ from ENCYCLOPEDIA OF TRIANGLE, therefore $P \in OI_A.$

Let $\omega$ be circumcircle of $\triangle ABC$ centered at $O.$ Let $Y_1,$ and $Z_1$ be crosspoints of $\omega$ and $BY,$ and $CZ,$ respectively. Let $T$ be crosspoint of $YZ$ and $Y_1 Z_1.$ In accordance the Pascal theorem for pentagon $AZ_1BCY_1,$ $AT$ is tangent to $\omega$ at $A.$

Let $I_A, I_B, I_C$ be $A, B,$ and $C$ -excenters of $\triangle ABC.$ Denote $a = BC, b = AC, c = AB, 2\alpha = \angle CAB, 2\beta = \angle ABC, 2\gamma = \angle ACB,$ $\psi = 90^\circ – \gamma + \beta, X = AI_A \cap \omega, X_1 = BC \cap AI_A,$ $I = BI_B \cap CI_C, U= YZ \cap AI_A, W = Y_1Z_1 \cap AI_A,$ $V$ is the foot ot perpendicular from $O$ to $AI_A.$

$I$ is ortocenter of $\triangle I_A I_B I_C$ and incenter of $\triangle ABC.$

$\omega$ is the Nine–point circle of $\triangle I_A I_B I_C.$

$Y_1$ is the midpoint of $II_B, Z_1$ is the midpoint of $II_C$ in accordance with property of Nine–point circle $\implies$ $Y_1Z_1 || I_B AI_C || VO, IW = AW \implies TW \perp AI.$ $\angle AXC = 180 ^\circ – 2\gamma – \alpha = 90 ^\circ – \gamma + \beta = \psi.$ $\angle TAI = \angle VOA = 2\beta + \alpha = 90 ^\circ – \gamma + \beta = \psi.$ $I_A X_1 = IX_1 = BX_1 = 2R \sin \alpha \implies$ $\cot \angle OI_A A = \frac {VI_A}{VO} = \frac {R \sin \psi + 2R \sin \alpha}{R \cos \psi} = \tan \psi + \frac{2 \sin\alpha}{\cos \psi}.$ $CX = \frac {ab}{b+c} \implies \frac {AI}{IX}= \frac {AC}{CX}= \frac {b+c}{a} \implies AI = AX \frac {b+c}{a+b+c},$ $AW = \frac {AI}{2}, UW = AU – AW,$ In $\triangle ABC$ segment $YZ$ cross segment $AX \implies \frac {AU}{UX} = \frac {m + nk}{k+1},$ where $n = \frac {a}{b}, m = \frac{a}{c}, k=\frac {b}{c},$ $\frac {AU}{UX} = \frac{2a}{b+c} \implies AU = AX \cdot \frac {b+c}{2a +b +c}.$ $\frac {AU – AW}{AW} = \frac {b+c} {2a + b + c}.$

$\cot \angle UTW = \frac {TW}{UW} = \frac {AW \cdot \tan \psi}{AU – AW} = \tan \psi \cdot \frac {2a +b+c}{b+c} =$ $=\tan \psi \cdot \frac {2a}{b+c} + \tan \psi = \frac {2 \sin \alpha}{\sin \psi} \tan \psi + \tan \psi = \tan \psi + \frac{2 \sin\alpha}{\cos \psi}$ $\implies \angle UTW = \angle OI_AA .$ $TW \perp AI_A \implies TYZ \perp OI_A.$

Let $\triangle II_B I_C$ be the base triangle with orthocenter $I_A,$ center of Nine-points circle $O \implies OI_A$ be the Euler line of $\triangle II_B I_C.$

$\triangle ABC$ is orthic triangle of $\triangle II_B I_C,$

$\triangle DEF$ is orthic-of-orthic triangle.

$P$ is perspector of base triangle and orthic-of-orthic triangle.

Therefore $P$ is point $X(24)$ of ENCYCLOPEDIA OF TRIANGLE CENTERS which lies on Euler line of the base triangle.

Claim $\frac {AU}{UX} = \frac {m + nk}{k + 1}.$ Proof $\frac {[AYZ]}{[ABC]} = \frac {AZ \cdot AY}{AB \cdot AC} = \frac {1}{(n + 1) \cdot (m+1)},$ $\frac {[BXZ]}{[ABC]} = \frac {BZ \cdot BX}{AB \cdot BC} = \frac {n}{(n + 1) \cdot (k+1)},$ $\frac {[CXY]}{[ABC]} = \frac {CY \cdot CX}{AC \cdot BC} = \frac {mk}{(m + 1) \cdot (k+1)},$ $\frac {[XYZ]}{[ABC]} = 1 - \frac {[AYZ]}{[ABC]} – \frac {[BXZ]}{[ABC]} – \frac {[CXY]}{[ABC]} = \frac {m+nk}{(m + 1) \cdot (k+1)\cdot (n+1)},$ $\frac {AU}{UX} = \frac {[AYZ]}{[XYZ]} = \frac {m + nk}{k + 1}.$

Kimberling point X(24)

$1.$ Perspector of Triangle $ABC$ and Orthic Triangle of the Orthic Triangle. Denote $T_0$ obtuse or acute $\triangle ABC.$ Let $T_0$ be the base triangle, $T_1 = \triangle DEF$ be Orthic triangle of $T_0, T_2 = \triangle UVW$ be Orthic Triangle of the Orthic Triangle of $T_0$ . Let $O$ and $H$ be the circumcenter and orthocenter of $T_0.$

Then $\triangle T_0$ and $\triangle T_2$ are homothetic, the point $P,$ center of this homothety lies on Euler line $OH$ of $T_0.$

The ratio of the homothety is $k = \frac {\vec {PH}}{\vec {OP}}= 4 \cos A \cos B \cos C.$

Proof

WLOG, we use case $\angle A = \alpha > 90^\circ.$ Let $B'$ be reflection $H$ in $DE.$

In accordance with Claim, $\angle BVD = \angle HVE \implies B', V,$ and $B$ are collinear.

Similarly, $C, W,$ and $C',$ were $C'$ is reflection $H$ in $DF,$ are collinear.

Denote $\angle ABC = \beta = \angle CHD, \angle ACB = \gamma = \angle BHD \implies$

$\angle HDF = \angle HDE = \angle DHB' = \angle DHC' = 180^\circ - \alpha.$

$B'C' \perp HD, BC \perp HD \implies BC|| B'C'.$ $OB = OC, HB' = HC', \angle BOC = \angle B'HC' = 360^\circ - 2 \alpha \implies OB ||HB', OC || HC' \implies$

$\triangle HB'C' \sim \triangle OBC, BB', CC'$ and $HO$ are concurrent at point $P.$

In accordance with Claim, $\angle HUF = \angle AUF \implies$ points $H$ and $P$ are isogonal conjugate with respect $\triangle UVW.$

$\angle HDE = \alpha - 90^\circ, \angle HCD = 90^\circ - \beta \implies$

$HB' = 2 HD \sin (\alpha - 90^\circ) = - 2 CD \tan(90^\circ- \beta) \cos \alpha = - 2 AC \cos \gamma \frac {\cos \beta}{\sin \beta} \cos \alpha = - 4 OB \cos A \cos B \cos C.$ $k = \frac {HB'}{OB} = \frac {HP}{OP}= - 4 \cos A \cos B \cos C \implies \frac {\vec {PH}}{\vec {OP}}= 4 \cos A \cos B \cos C.$

Claim

Let $\triangle ABC$ be an acute triangle, and let $AH, BD',$ and $CD$ denote its altitudes. Lines $DD'$ and $BC$ meet at $Q, HS \perp DD'.$ Prove that $\angle BSH = \angle CSH.$

Proof

Let $\omega$ be the circle $BCD'D$ centered at $O (O$ is midpoint $BC).$

Let $\omega$ meet $AH$ at $P.$ Let $\Omega$ be the circle centered at $Q$ with radius $QP.$

Let $\Theta$ be the circle with diameter $OQ.$

We know that $OB = OP = OC = R, PH^2 = R^2 – OH^2 \implies$ $QP^2 + R^2 = (QH+ HO)^2 \implies P \in \Theta, \Omega \perp \omega.$

Let $I_{\Omega}$ be inversion with respect $\Omega, I_{\Omega}(B) = C.$

Denote $I_{\Omega}(D) = D', I_{\Omega}(S) = S',$ $QH \cdot QO = QP^2 \implies I_{\Omega}(H) = O.$ $HS \perp DD' \implies S'O \perp BC \implies BS' = CS' \implies \angle OCS' = \angle OBS'.$

$\angle QSB = \angle QCS' = \angle OCS' = \angle OBS' = \angle CSS'.$

$\angle BSH = 90 ^\circ – \angle QSB = 90 ^\circ – \angle CSS' =\angle CSH.$

vladimir.shelomovskii@gmail.com, vvsss

2016 USAMO (Problems • Resources)
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2016 USAMO Problems/Problem 3

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Solution

Solution 2

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