Reflections of an interior point and a surprising collinearity

empty_estus·May 9, 2025

I have several equal angles from reflecting P in the sides, but I can't connect the lines MD and LE to the line AK. Would an inversion about A simplify the two second intersections, or is there a direct spiral-similarity configuration? I would appreciate a construction hint before a full solution. Problem: Let

\triangle ABC

be an acute triangle and

P

be an inner point of

\triangle ABC

. Let

K

L

and

M

be the reflections of

P

across

BC

AC

and

AB

, respectively. Let

D

and

E

be the second points of intersection of

\odot(PBC)

with lines

AB

and

AC

, respectively. Let lines

MD

and

LE

intersect at

F

. Prove that

F

A

and

K

are collinear. Source: Baltic Way 2023 Shortlist · MNET-01J6

Open MNET-01J6 in the question bank →

Reflections of an interior point and a surprising collinearity

empty_estus·May 9, 2025

\triangle ABC

be an acute triangle and

P

be an inner point of

\triangle ABC

. Let

K

L

and

M

be the reflections of

P

across

BC

AC

and

AB

, respectively. Let

D

and

E

be the second points of intersection of

\odot(PBC)

with lines

AB

and

AC

, respectively. Let lines

MD

and

LE

intersect at

F

. Prove that

F

A

and

K

are collinear. Source: Baltic Way 2023 Shortlist · MNET-01J6

Open MNET-01J6 in the question bank →

Reflections of an interior point and a surprising collinearity

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Reflections of an interior point and a surprising collinearity

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