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 1 of 5: Show nesting and convergence

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P_nP_{n+1}P_{n+2}\subset P_{n-1}P_nP_{n+1}

Detailed analysis

The perpendicular-foot construction puts each triangle $P_nP_{n+1}P_{n+2}$ inside the preceding triangle $P_{n-1}P_nP_{n+1}$ . All these triangles are similar, while their sizes decrease to zero. The nested closed triangles therefore have one common point $P$ , and $P_n\to P$ .