1. ## Fleas with Triangles

a) In one of your worst nightmares you are a contestant on a TV game show that is designed to make you look silly. You and two other contestants have been placed at the mid-points of the edges of a triangle (one contestant per edge). Each contestant has a pet flea that is sitting on the opposing vertex. These are specially trained homing fleas, once released from their vertex prisons they will travel in a straight line to their loving owners (i.e., you). The winner is the person who is first re-united with their pet flea. Your triangle has sides of length 6, 9 and 11 metres and your pet fleas are in top shape having been recently measured (with a flea-radar gun) to run at a staggering 0.666 m/sec. You are on the side of length 9 metres. All 3 fleas are released at the same time. Are you a winner?

(b) The fleas have been equipped with special little boots that leave a distinct trail
as they run home (to you). Thus each flea draws a line over the triangle. What
interesting fact can you deduce about the three lines drawn by the
fleas? You can assume here that to the human eye the lines are continuous (in reality they are dotted lines, one dot for each boot print, but the dots are so small and so close we don't notice the individual dots seeing instead a steady straight line).

2. ## Re: Fleas with Triangles

Hello, bakchormee!

a) In one of your worst nightmares you are a contestant on a TV game show
that is designed to make you look silly. You and two other contestants have been
placed at the mid-points of the edges of a triangle. Each contestant has a pet flea
that is sitting on the opposing vertex. These are specially trained homing fleas;
once released from their vertex prisons they will travel in a straight line to their owners.
The winner is the person who is first re-united with their pet flea. The triangle has
sides of length 6, 9 and 11 metres and the pet fleas are in top shape having been
recently measured (with a flea-radar gun) to run at a staggering 0.666 m/sec.
You are on the side of length 9 metres. . All 3 fleas are released at the same time.
Are you a winner?

No, the winner is the person at the midpoint of 11-meter side.

. . [[The longest side is closest to its opposite vertex.]

(b) The fleas have been equipped with special little boots that leave a distinct trail
as they run home (to you). .Thus each flea draws a line over the triangle.
What interesting fact can you deduce about the three lines drawn by the fleas?

The fleas run along the medians of the triangle.
Hence, their paths intersect at a single point, the centroid
. . (center of gravity) of the triangle.