2013 APMO Problems/Problem 5

Problem

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ , and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$ . The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$ . Let $E$ be the second point of intersection between $AQ$ and $\omega$ . Prove that $B$ , $E$ , $R$ are collinear.

Solution

Solution 1

Let $R'=AD \cap BE$ . Note that the tangents at $B$ and $D$ concur on $AC$ at $P$ , so $ABCD$ is harmonic, hence the tangents at $C$ and $A$ concur on $BD$ at $X$ , say.

Now apply Pascal's Theorem to hexagon $AAEBDD$ to find that $X=AA \cap BD$ , $Q=AE \cap DD$ and $R'=BE \cap AD$ are collinear. Now note that $X$ and $Q$ both lie on the tangent at $C$ , hence $R'$ also lies on the tangent at $C$ . It follows that $R'=CC \cap AD \cap BE=CC \cap AD=R$ . So $R'$ and $R$ are in fact the same point. Since $R'=R$ lies on $BE$ by definition, it follows that $B, E, R$ , are indeed collinear, and thus the problem is solved.

Solution 2

We use complex numbers. Let $\omega$ be the unit circle, and let the lowercase letter of a point be its complex coordinate.

Since $P$ lies on the intersection of the tangents to $\omega$ at $B$ and $D$ , we have $p = \frac{2bd}{b+d}$ . In addition, $P$ lies on chord $AC$ , so $p = a+c-ac\bar{p}$ . This implies that $\frac{2bd}{b+d} = a+c - \frac{2ac}{b+d}$ , or $ab+bc+cd+da=2ac+2bd$ .

$Q$ lies on the tangent at $C$ , and lies on $PD$ , so $q = \frac{2cd}{c+d}$ .

$R$ lies on chord $AD$ and on the tangent at $C$ . Therefore we have $r = 2c - c^2\bar{r}$ and $r = a+d-ad\bar{r}$ . Solving for $r$ yields $r = a + d - ad \frac{2c-r}{c^2} \implies r = \frac{ac^2+c^2d-2acd}{c^2-ad}$ $A, Q, E$ are collinear, so we have $\frac{a-q}{\bar{a}-\bar{q}} = \frac{a-e}{\bar{a}-\bar{e}}$ , or $\frac{a-\frac{2cd}{c+d}}{\frac{1}{a}-\frac{2}{c+d}} = -ae \implies e = \frac{ac + ad - 2cd}{2a-c-d}$ We must prove that $B, E, R$ are collinear, or that $\frac{b-e}{\bar{b}-\bar{e}} = \frac{b-r}{\bar{b}-\bar{r}} \implies -be = \frac{b-r}{\frac{1}{b} - \bar{r}}$ or \begin{align*}\frac{abc+abd-2bcd}{c+d-2a} = \frac{b-\frac{ac^2+c^2d-2acd}{c^2-ad}}{\frac{1}{b}- \frac{\frac{1}{ac^2}+\frac{1}{c^2d}-\frac{2}{acd}}{\frac{1}{c^2}-\frac{1}{ad}}} = \frac{b - \frac{ac^2+c^2d-2acd}{c^2-ad}}{\frac{1}{b} - \frac{d+a-2c}{ad-c^2}} = \frac{b(c^2-ad)-(ac^2+c^2d-2acd)}{\frac{c^2-ad}{b} + (d+a-2c)}\end{align*} Cross-multiplying, we have \begin{align*}(ac+ad-2cd)(c^2-ad+bd+ab-2bc) = (c+d-2a)(bc^2-abd - ac^2 - c^2d + 2acd) \\ \implies (a-c)(c-d)(ab+bc+cd+da-2ac-2bd) = 0\end{align*} which is true.

Solution 3

Set $T = \overline{BD} \cap \overline{CR}$ , $K = \overline{AC} \cap \overline{BD}$ , $Z = \overline{AB} \cap \overline{CR}$ and $E' = \overline{BR} \cap \omega$ , where $E' \neq A$ . Note that since $ABCD$ is harmonic, we have $T,K,B,D$ collinear and with $-1 = (T,K;B,D) \stackrel{A}{=} (T,C; Z,R) \stackrel{B}{=} (D,C; A,E').$ But $DACE$ is harmonic; therefore $E = E'$ . $\blacksquare$

Solution 4

Use barycentrics on $CBD$ , in that order. It is easy to derive that $P=(-a^2:b^2:c^2)$ and $A=(-a^2:2b^2:2c^2)$ . Clearly, line $PD$ has equation $\frac{x}{y}=\frac{-a^2}{b^2}$ , and the tangent from $C$ has equation $c^2y+b^2z=0$ , so we get that $Q=(-a^2:b^2:c^2)$ . Line $AD$ has equation $\frac{x}{y}=\frac{-a^2}{2b^2}$ , so we also get that $R=(-a^2:2b^2:-2c^2)$ . Line $AQ$ has equation $4b^2c^2x+3a^2c^2y-a^2b^2z=0$ , and line $BR$ has equation $\frac{x}{z}=\frac{a^2}{2c^2}$ , so we quickly derive that $E$ has coordinates $(3a^2:-2b^2:6c^2)$ , and it is easy to verify that this lies on the circumcircle $\displaystyle{\sum_{\text{cyc}}a^2yz=0}$ , so we're done.

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