MathDB

Let

ABC

be a triangle,

O

its circumcenter and

R

the radius of its circumcircle.Denote by

O_{1}

the symmetric of

O

with respect to

BC

O_{2}

the symmetric of

O

with respect to

AC

and by

O_{3}

the symmetric of

O

with respect to

AB

. (a)Prove that the circles

C_{1}(O_{1},R)

C_{2}(O_{2},R)

C_{3}(O_{3},R)

have a common point. (b)Denote by

T

this point.Let

l

be an arbitary line passing through

T

which intersects

C_{1}

L

C_{2}

M

and

C_{3}

K

.From

K,L,M

drop perpendiculars to

AB,BC,AC

respectively.Prove that these perpendiculars pass through a point.

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