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rioplatense locus with circle, diameter, secant

Source: Rioplatense Olympiad 1996 level 3 P4

September 6, 2018

geometrycircleLocus

Problem Statement

Let

S

be the circle of center

O

and radius

R

, and let

A, A'

be two diametrically opposite points in

S

. Let

P

be the midpoint of

OA'

and

\ell

a line passing through

P

, different from

AA '

and from the perpendicular on

AA '

. Let

B

and

C

be the intersection points of

\ell

with

S

and let

M

be the midpoint of

BC

. a) Let

H

be the foot of the altitude from

A

in the triangle

ABC

. Let

D

be the intersection point of the line

A'M

with

AH

. Determine the locus of point

D

while

\ell

varies . b) Line

AM

intersects

OD

at

I

. Prove that

2 OI = ID

and determine the locus of point

I

while

\ell

varies .

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