MathDB

Moving a point in the plane and magical fixed points

Source: RMM 2024 Problem 5

February 29, 2024

geometryfixed pointssymmetrymoving pointsRMMrmm 2024

Problem Statement

Let

BC

be a fixed segment in the plane, and let

A

be a variable point in the plane not on the line

BC

. Distinct points

X

and

Y

are chosen on the rays

CA^\to

and

BA^\to

, respectively, such that

\angle CBX = \angle YCB = \angle BAC

. Assume that the tangents to the circumcircle of

ABC

at

B

and

C

meet line

XY

at

P

and

Q

, respectively, such that the points

X

,

P

,

Y

and

Q

are pairwise distinct and lie on the same side of

BC

. Let

\Omega_1

be the circle through

X

and

P

centred on

BC

. Similarly, let

\Omega_2

be the circle through

Y

and

Q

centred on

BC

. Prove that

\Omega_1

and

\Omega_2

intersect at two fixed points as

A

varies.
Daniel Pham Nguyen, Denmark

Back to Problems View on AoPS