Motorola M500 Bedienungsanleitung PDF herunterladen (Seite 16)

Page 14 M500 180

Solution 178.2 – Construct another square

Given a unit circle and its centre, construct a square of side one

unit, using only a ruler.

Dick Boardman

My solution uses the following construction: Given a line segment AB, its

midpoint O and a point P , draw a line through P parallel to AB.

Draw line AP and extend it to an arbitrary point C. Draw lines OC,

BC and BP. Let BP and OC intersect at E. Draw AE and extend it to

meet BC at D. Then PD is parallel to AOB.

Proof. Area of triangle ACO = area of triangle BCO. Area of triangle

AEO = area of triangle BEO. By subtraction,

area of triangle AEC = area of triangle BEC. (1)

Let CP/AP = m. Area of triangle CBP = m times area of triangle ABP.

Area of triangle CEP = m times area of triangle AEP. By subtraction,

area of triangle CEB = m times area of triangle AEB. (2)

Using (1) and (2),

area of triangle AEC = m times area of triangle AEB. (3)

Let CD/DB = n. Then similarly

area of triangle AEC = n times area of triangle AEB. (4)

From (3) and (4) we conclude that m = n and so P D is parallel to AB.

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Motorola M500 Bedienungsanleitung Seite 16