RMO is really insane

Geometry Level 3

Let $ABC$ be a triangle.Let $D$ and $C$ be points on segments $BC$ such that $BD=DE=EC$ .

Let $F$ be the midpoint of $AC$ . Let BF intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of area of the triangle $APQ$ to that of the quadrilateral $PDEQ$ .

\frac {8}{10}

\frac {9}{10}

\frac {8}{11}

\frac {9} {11}

1 solution

Ayush Pattnayak
Mar 3, 2016

Let x=[APQ] and y=[PQED]. Let A represent area of [ADE]. x+y=A. Apply Menelaus' theorem in triangle ABC to get AP=3PD. So AP=3AD/4 . Again use the theorem in triangle AEC and we get AQ/QE=3/2. So,AQ=3AE/5. Now, x=3AD/4 x 3AE/5 x sin(angleA) = 9A/20 .So, y=11A/20. Ratio is --》 9:11.

use latex please

SRIJAN Singh - 8 months, 3 weeks ago

RMO is really insane

1 solution

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