MathDB

Let

AA_1, BB_1

and

CC_1

be the altitudes of an acute-angled triangle

ABC.

AA_1

meets

B_1C_1

in a point

K.

The circumcircles of triangles

A_1KC_1

and

A_1KB_1

intersect the lines

AB

and

AC

for the second time at points

N

and

L

respectively. Prove that
a) The sum of diameters of these two circles is equal to

BC,

\frac{A_1N}{BB_1} + \frac{A_1L}{CC_1}=1.

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