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Geometrical inequality

Source: Romania EGMO TST 2023 Day 1 P3

February 23, 2023

geometryinequalitiesromania

Problem Statement

Let

D{}

be a point inside the triangle

ABC

. Let

E{}

and

F{}

be the projections of

D{}

onto

AB

and

AC

, respectively. The lines

BD

and

CD

intersect the circumcircle of

ABC

the second time at

M{}

and

N{}

, respectively. Prove that

\frac{EF}{MN}\geqslant \frac{r}{R},

where

r{}

and

R{}

are the inradius and circumradius of

ABC

, respectively.

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