MathDB

A semicircular arc and a diameter

AB

with a length of

2

are given. Let

O

be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point

P

at the distance

d

from the midpoint of the diameter

O

0 <d <1

. A line through

A

and

P

intersects the semicircle at point

C

. Through point

P

we draw another line at right angle against

AC

that intersects the semicircle at point

D

. Through point

C

we draw a line

l_1

, parallel to

PD

and then a line

l_2

, through

D

parallel to

PC

. The lines

l_1

and

l_2

intersect at point

E

. Show that the distance between

O

and

E

is equal to

\sqrt{2- d^2}

5