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Geometry Mathley 12.2 concurrency, fixed line

Source:

June 7, 2020

geometryconcurrentfixedfixed line

Problem Statement

Let

K

be the midpoint of a fixed line segment

AB

, two circles

(O)

and

(O')

with variable radius each such that the straight line

OO'

is throughK and

K

is inside

(O)

, the two circles meet at

A

and

C

, center

O'

is on the circumference of

(O)

and

O

is interior to

(O')

. Assume that

M

is the midpoint of

AC, H

the projection of

C

onto the perpendicular bisector of segment

AB

. Let

I

be a variable point on the arc

AC

of circle

(O')

that is inside

(O), I

is not on the line

OO'

. Let

J

be the reflection of

I

about

O

. The tangent of

(O')

at

I

meets

AC

at

N

. Circle

(O'JN)

meets

IJ

at

P

, distinct from

J

, circle

(OMP)

intersects

MI

at

Q

distinct from

M

. Prove that (a) the intersection of

PQ

and

O'I

is on the circumference of

(O)

. (b) there exist a location of

I

such that the line segment

AI

meets

(O)

at

R

and the straight line

BI

meets

(O')

at

S

, then the lines

AS

and

KR

meets at a point on the circumference of

(O)

. (c) the intersection

G

of lines

KC

and

HB

moves on a fixed line.
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