MathDB

Goes through fixed points

Source: Vietnam TST 2021 P5

April 2, 2021

geometry

Problem Statement

Given a fixed circle

(O)

and two fixed points

B, C

on that circle, let

A

be a moving point on

(O)

such that

\triangle ABC

is acute and scalene. Let

I

be the midpoint of

BC

and let

AD, BE, CF

be the three heights of

\triangle ABC

. In two rays

\overrightarrow{FA}, \overrightarrow{EA}

, we pick respectively

M,N

such that

FM = CE, EN = BF

. Let

L

be the intersection of

MN

and

EF

, and let

G \neq L

be the second intersection of

(LEN)

and

(LFM)

.
a) Show that the circle

(MNG)

always goes through a fixed point.
b) Let

AD

intersects

(O)

at

K \neq A

. In the tangent line through

D

of

(DKI)

, we pick

P,Q

such that

GP \parallel AB, GQ \parallel AC

. Let

T

be the center of

(GPQ)

. Show that

GT

always goes through a fixed point.

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