Problem 4

Let

P_1

and

P_3

be fixed points. Let

P_2

lie on the line through

P_3

perpendicular to

P_1P_3

. Define

P_{n+1}

as the foot of the perpendicular from

P_n

P_{n-1}P_{n-2}

. Show that the sequence converges to a point

P

, and determine the locus of

P

P_2

varies.

Step 2 of 5: Use the first similarity

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P_5P_4P_6\sim P_1P_2P_3,\qquad P_4P_5\parallel P_1P_2

Detailed analysis

The triangle $P_5P_4P_6$ is similar, in that vertex order, to $P_1P_2P_3$ . Since $P_4P_5\parallel P_1P_2$ , the similarity is a half-turn about $P$ . Hence $P_1,P,P_5$ are collinear.