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Intersection of Simson lines lies on NPC

Source: 2022 China TST, Test 2, P2

March 28, 2022

geometrySimson lineNine Point Circle

Problem Statement

Given a non-right triangle

ABC

with

BC>AC>AB

. Two points

P_1 \neq P_2

on the plane satisfy that, for

i=1,2

, if

D_i, E_i

and

CP_i

intersect the circumcircle of the triangle

ABC

at

D_i, E_i

, and

F_i

, respectively, then

D_iE_i \perp D_iF_i

and

D_iE_i = D_iF_i \neq 0

. Let the line

P_1P_2

intersects the circumcircle of

ABC

at

Q_1

and

Q_2

. The Simson lines of

Q_1

,

Q_2

with respect to

ABC

intersect at

W

.
Prove that

W

lies on the nine-point circle of

ABC

.

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