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IMO Long List 1986 Geometry problem 50

Source:

August 29, 2010

geometryincenterperimetergeometry unsolved

Problem Statement

Let

D

be the point on the side

BC

of the triangle

ABC

such that

AD

is the bisector of

\angle CAB

. Let

I

be the incenter of

ABC.

(a) Construct the points

P

and

Q

on the sides

AB

and

AC

, respectively, such that

PQ

is parallel to

BC

and the perimeter of the triangle

APQ

is equal to

k \cdot BC

, where

k

is a given rational number.
(b) Let

R

be the intersection point of

PQ

and

AD

. For what value of

k

does the equality

AR = RI

hold?
(c) In which case do the equalities

AR = RI = ID

hold?

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