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 4 of 5: Bound the position on the circle

Previous step Next step

P_3P_5=\frac12P_1P_3\sin2\angle P_1

Detailed analysis

Writing the relevant angles of $P_1P_2P_3$ as $\angle P_1$ and $\angle P_2$ , the construction gives $P_3P_5=P_3P_1\sin\angle P_1\cos\angle P_2=(1/2)P_3P_1\sin2\angle P_1$ . Hence $|P_3P_5|\le P_1P_3/2$ ; both signs occur as $P_2$ moves on either side of $P_3$ .