Question 4
Triangle $A_1A_2A_3$ is right-angled at $A_3$. For $n\ge3$, draw the perpendicular from $A_n$ to the line $A_{n-2}A_{n-1}$, meeting it at $A_{n+1}$.
(a) Prove that if this process continues indefinitely, exactly one point $P$ is interior to every triangle $A_{n-2}A_{n-1}A_n$.
(b) With $A_1,A_3$ fixed, let $A_2$ vary over all permissible plane positions. Find the locus of $P$.
(a) Prove that if this process continues indefinitely, exactly one point $P$ is interior to every triangle $A_{n-2}A_{n-1}A_n$.
(b) With $A_1,A_3$ fixed, let $A_2$ vary over all permissible plane positions. Find the locus of $P$.