Capture the Flag

By:

Allison Kootsikas

Three teams, A, B, and C, each start from a vertex of a scalene triangular field. Their goal is to be the first team to grab the flag that is located inside the triangular field. If the game is fair, then each team has to run the same distance to get to the flag.

Where should the flag be positioned for the game to be fair? Describe how you found the position.

For this problem, I have to find where the flag needs to be positioned so that it is equidistant to each of the vertices.  I used GSP for this investigation.

Before getting to the flag, I had to investigate a few other roads to the solution to this problem.  The first thing I decided to find out was the intersection of the altitudes.  I drew a GSP sketch to do this.  I drew the altitudes as perpendicular lines to the opposite sides.

I found that the point of intersection was outside of the triangle.  The intersection is called the orthocenter.  I then drew a circle from the orthocenter to a vertex to see if the orthocenter would be equidistant from all of the vertices and found this was not the case.

The next path on the road to the solution I explored was to find the intersection of the angle bisectors.   I drew in the angle bisectors of the triangle.  I then found the intersection point, called the incenter.

The incenter was not equidistant to the vertices.  I prove this by drawing a circle with the center at the incenter and the radius meeting a vertex.

The incenter was not equidistant to each of the vertices.  The only time it would be would be for an equilateral triangle, which does not fit the requirements of the problem.  You can see a sketch of the equilateral triangle here.

I then decided to try to find the intersection of the midpoints.  I drew in the midpoints and connected them.

I then drew in segments from the midpoint to the opposite vertex to find the point of intersection of the midpoints.

This is called the centroid of the triangle.

To check to see if the centroid is equidistant from the vertices, I drew a circle in with the center being the centroid and the radius going to a vertex.

When the circle is drawn, the centroid is not equidistant from the vertices in a scalene triangle.

I then decided to find the intersection of the perpendicular bisectors.  I started by drawing in the midpoints and then drawing a line perpendicular to the line.   The intersection of these points is called the circumcenter.

Next, I drew segments from the vertices to the circumcenter.

The next step was to draw in a circle to check and see if the circumcenter was equidistant from the vertices.  The radius is drawn from the circumcenter to the vertex.  This circle is called a circumcircle.

The circumcircle is equidistant to the vertices.

EUREKA!  The game is fair if the method used to find the location of the flag is to find the circumcenter of the triangle.