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Midpoints on square sequence

Author:: Juan Carlos Ponce Campuzano

Topic:: Geometry, Square

Construction:

Starting with the vertices

of a square, we construct further points as follows:

is the midpoint of ,
is the midpoint of ,
is the midpoint of , and so on.

The polygonal spiral path

approaches a point

inside the square, as shown in the animation below. You can pause the animation and activate the box Value to analyse the values of

.

1. If the coordinates of

are

, show that

and find a similar equation for the

-coordinates. 2. Find the coordinates of

.

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