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Grasshopper jumps in an equilateral triangle

Source: Pan African 2001

October 4, 2005

geometrygeometric transformation

Problem Statement

Let

ABC

be an equilateral triangle and let

P_0

be a point outside this triangle, such that

\triangle{AP_0C}

is an isoscele triangle with a right angle at

P_0

. A grasshopper starts from

P_0

and turns around the triangle as follows. From

P_0

the grasshopper jumps to

P_1

, which is the symmetric point of

P_0

with respect to

A

. From

P_1

, the grasshopper jumps to

P_2

, which is the symmetric point of

P_1

with respect to

B

. Then the grasshopper jumps to

P_3

which is the symmetric point of

P_2

with respect to

C

, and so on. Compare the distance

P_0P_1

and

P_0P_n

.

n \in N

.

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