2019 USAJMO Problems/Problem 3

Problem

Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD=\angle BPC$. Show that line $PE$ bisects $\overline{CD}$.

Solution 1

Let $PE \cap DC = M$. Also, let $N$ be the midpoint of $AB$.

Note that only one point $P$ satisfies the given angle condition. With this in mind, construct $P'$ with the following properties:

$AP' \cdot AB = AD^2 \quad \text{and} \quad BP' \cdot AB = CD^2$

Claim: $P = P'$

Proof:

The conditions imply the similarities $ADP \sim ABD$ and $BCP \sim BAC$ whence $\measuredangle APD = \measuredangle BDA = \measuredangle BCA = \measuredangle CPB$ as desired. $\square$

Claim: $PE$ is a symmedian in $AEB$

Proof:

We have

\[AP \cdot AB = AD^2 \iff AB^2 \cdot AP = AD^2 \cdot AB\] \[\iff \left( \frac{AB}{AD} \right)^2 = \frac{AB}{AP}\] \[\iff \left( \frac{AB}{AD} \right)^2 - 1 = \frac{AB}{AP} - 1\] \[\iff \frac{AB^2 - AD^2}{AD^2} = \frac{BP}{AP}\] \[\iff \left(\frac{BC}{AD} \right)^2 = \left(\frac{BE}{AE} \right)^2 = \frac{BP}{AP}\]

as desired. $\square$

Since $P$ is the isogonal conjugate of $N$, $\measuredangle PEA = \measuredangle MEC = \measuredangle BEN$. However $\measuredangle MEC = \measuredangle BEN$ implies that $M$ is the midpoint of $CD$ from similar triangles, so we are done. $\square$

~sriraamster

Solution 2

By monoticity, we can see that the point $P$ is unique. Therefore, if we find another point $P'$ with all the same properties as $P$, then $P = P'$

Part 1) Let $N$ be a point on $\overline{AB}$ such that $AN\cdot AB = AD^2$, and $BN \cdot AB = BC^2$. Obviously $N$ exists because adding the two equations gives $AN\cdot AB + BN \cdot AB = AD^2 + BC^2 = AB^2$, which is the problem statement. Notice that converse PoP gives\[AN\cdot AB = AD^2 \implies \bigtriangleup ADN \sim \bigtriangleup ABD\]\[BN\cdot AB = AD^2 \implies \bigtriangleup CBN \sim \bigtriangleup ABC\]Therefore, $\angle AND = \angle ADB = \angle ACB = \angle CNB$, so $N$ does indeed satisfy all the conditions $P$ does, so $N = P$. Hence, $\bigtriangleup ADP \sim \bigtriangleup ABD$ and $\bigtriangleup CBP \sim\bigtriangleup ABC$.

Part 2) Define $G$ as the midpoint of $CD$. Furthermore, create a point $X$ such that $DX || DC$ and $CX || ED$. Obviously $XCED$ must be a parallelogram. Now we set up for Jacobi's. The problem already gives us that $\angle APD = \angle CPB$, which is good for starters. Furthermore, $\bigtriangleup ADP \sim \bigtriangleup ABD$ tells us that\[\angle ADP = \angle ABD = \angle ACD = \angle XDC\]This gives us our second needed angle equivalence. Lastly, $\bigtriangleup CBP \sim\bigtriangleup ABC$ will give\[\angle BCP = \angle BAC = \angle BDC = \angle XCD\]which is our last necessary angle equivalence to apply Jacobi's. Finally, applying Jacobi's tells us that $AC$, $BD$, and $XP$ are concurrent $\implies$ $X$, $E$, $P$ collinear. Additionally, since parallelogram diagonals bisect each other, $X$, $G$, and $E$ are collinear, so finally we obtain that $P$, $E$, and $G$ are collinear, as desired.

-jj_ca888

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See also

2019 USAJMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6
All USAJMO Problems and Solutions
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