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 5 of 5: Describe the locus arc

Previous step Next step

\angle XP_1Y=2\tan^{-1}(1/2),\qquad\angle XOY=4\tan^{-1}(1/2)

Detailed analysis

Therefore the locus is the arc $XP_3Y$ of the circle with diameter $P_1P_3$ , with $XP_3=YP_3$ and $\angle XP_1Y=2\tan^{-1}(1/2)$ . If $O$ is the midpoint of $P_1P_3$ , its central angle is $\angle XOY=4\tan^{-1}(1/2)$ .