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 3 of 5: Locate

P

on a circle

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P_3P_4P_5\sim P_1P_2P_3,\qquad P_1P\perp P_3P

Detailed analysis

The triangle $P_3P_4P_5$ is obtained from $P_1P_2P_3$ by a quarter-turn about $P$ . It sends the line $P_1P_5$ to the perpendicular through $P_3$ , so $P_1P\perp P_3P$ . Thus $P$ lies on the circle with diameter $P_1P_3$ .