MathDB

concurrent wanted, circumcircle and touchpoints of incircle related

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p2

November 13, 2022

geometryincircleconcurrencyconcurrent

Problem Statement

Let

k

be the circumcircle of the acute triangle

ABC

. Its inscribed circle touches sides

BC

CA

and

AB

at points

D, E

and

F

respectively. The line

ED

intersects

k

at the points

M

and

N

, so that

E

lies between

M

and

D

. Let

K

and

L

be the second intersection points of the lines

NF

and

MF

respectively with

k

. Let

AK \cap BL = Q

. Prove that the lines

AL

BK

and

QF

intersect at a point.