MathDB

Joaquim, José and João participate of the worship of triangle

ABC

. It is well known that

ABC

is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to

ABC

, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice

A

, José vertice

B

and João will represent vertice

C

. Thus, they must form a triangle which is similar to

ABC

, in this order.
Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant

t

, Joaquim, for example, will move with constant velocity

v

to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to

ABC

in the desired order. The other participants act in the same way. If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal.
Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane

Problem Statement