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Concurrent

Source: Romania District Olympiad 2013,grade IX,(problem 2)

March 14, 2013

trigonometryratiogeometry proposedgeometry

Problem Statement

Given triangle

ABC

and the points

D,E\in \left( BC \right)

,

F,G\in \left( CA \right)

,

H,I\in \left( AB \right)

so that

BD=CE

,

CF=AG

and

AH=BI

. Note with

M,N,P

the midpoints of

\left[ GH \right]

,

\left[ DI \right]

and

\left[ EF \right]

and with

{M}'

the intersection of the segments

AM

and

BC

. a) Prove that

\frac{B{M}'}{C{M}'}=\frac{AG}{AH}\cdot \frac{AB}{AC}

. b) Prove that the segments

AM

,

BN

and

CP

are concurrent.

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