PQRE area wanted

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Source: 2018 Taiwan APMO preliminary

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#1 h Nov 10, 2020, 2:50 PM

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Let $ABCD$ be an unit aquare. $E,F$ be the midpoints of $CD,BC$ respectively. $AE$ intersects the diagonal $BD$ at $P$ . $AF$ intersects $BD,BE$ at $Q,R$ respectively. Find the area of quadrilateral $PQRE$ .

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#2 h Nov 10, 2020, 2:50 PM • 1 Y

Y by Ha_ha_ha

Ans: $\dfrac{2}{15}$

This post has been edited 1 time. Last edited by hsiangshen, Nov 10, 2020, 2:50 PM

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#5 h Nov 10, 2020, 9:06 PM • 2 Y

Y by Ha_ha_ha, Mango247

We use Cartesian coordinates. Let $D$ be the origin, $A=(0,1),C=(1,0),B=(1,1)$ . We have $E=(\tfrac 12,0)$ and $F=(1,\tfrac 12)$ . $\overline{BD}$ is governed by $y=x$ . $\overline{BE}$ is governed by $y=2x-1$ and $\overline{AF}$ is governed by $y=-\frac 12x+1$ . Now we can easily compute $P=(\tfrac 13,\tfrac 13),Q=(\tfrac 23,\tfrac 23),,R=(\tfrac 45,\tfrac 35)$ . We finish with Shoelace.

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#6 h Nov 10, 2020, 9:15 PM • 2 Y

Y by Ha_ha_ha, Mango247

We can also use Barycentric coordinates w.r.t. $\triangle DBC$ . From above, we have that $DP=PQ=QB$ . So we can immediately compute $D=(1,0,0),B=(0,1,0),C=(0,0,1),E=(\tfrac 12,0,\tfrac 12),F=(0,\tfrac 12,\tfrac12),P=(\tfrac 23,\tfrac 13,0),Q=(\tfrac 13,\tfrac23,0)$ . The coordinates of $R$ are also easy enough to compute with equations of lines $\overline{QF}\cap\overline{BE}=R$ . Then just compute $1-([DPE]+[BEC]+[BQR])$ with barycentric area formula.

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