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construction, locus, starting with perpendicular diameteres and tangents

Source: Vietnamese MO (VMO) 1970

August 23, 2018

geometryLocusconstructiontangentdiameter

Problem Statement

AB

and

CD

are perpendicular diameters of a circle.

L

is the tangent to the circle at

A

.

M

is a variable point on the minor arc

AC

. The ray

BM, DM

meet the line

L

at

P

and

Q

respectively. Show that

AP\cdot AQ = AB\cdot PQ

. Show how to construct the point

M

which gives

BQ

parallel to

DP

. If the lines

OP

and

BQ

meet at

N

find the locus of

N

. The lines

BP

and

BQ

meet the tangent at

D

at

P'

and

Q'

respectively. Find the relation between

P'

and

Q

'. The lines

D

P and

DQ

meet the line

BC

at

P"

and

Q"

respectively. Find the relation between

P"

and

Q"

.

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