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starting with a circle with diameter the centres of 2 externally tangent circles

Source: Irmo 2015 p1 q4

September 16, 2018

geometrycirclestangent

Problem Statement

Two circles

C_1

and

C_2

, with centres at

D

and

E

respectively, touch at

B

. The circle having

DE

as diameter intersects the circle

DE

at

H

and the circle

C_2

at

K

. The points

H

and

K

both lie on the same side of the line

DE

.

HK

extended in both directions meets the circle

C_1

at

L

and meets the circle

C_2

at

M

. Prove that (a)

|LH| = |KM|

(b) the line through

B

perpendicular to

DE

bisects

HK

.

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