I stood staring in disbelief as the ghost point began to move differently. I recreated this movement as well on GSP to be able to analyze what point it could be.
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Immediately I noticed something really nice about the movement of this ghost point. If I continue the line the ghost point is following down to the base of the triangle, we can see that the line goes through the midpoint of the base. Again, using my handy dandy protractor I measure the line compared to the base and find that the line of the ghost point is perpendicular to the base. So this line if extended to the base goes through the midpoint and is perpendicular. That would mean that we are seeing the trajectory of the circumcenter. The circumcenter is the point of concurrency for the perpendicular bisectors of the triangle.
I said I was ready for one more, and my Grandpa delivered a new puzzle for me.
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This one was a bit odd. I thought it looked like an eyebrow when I first saw it. I got out of that frame of mind and started focusing on what we could tell from the movement of the ghost point. The point seems to be moving closer and closer to the vertices but it never touches. In addition the ghost point always seems to be remaining inside of the triangle. I decided to draw a segment from the pocket watch to the ghost point and from the eyeglasses to the ghost point as well. With this I was able to find using my handy dandy protractor that the segments I drew in bisected the angle from the chalk to the pocket watch to the eyeglasses. That let me know the ghost point was following the trajectory of the point called the incenter. The incenter is the point of concurrency of the angle bisectors of a triangle. The incenter is the also the center for a circle that will be inside of the triangle and tangent to the three sides.
I let me Grandpa know of my discovery, when I heard the door open behind me and my Grandma come in. She asked me if I had enjoyed myself and I said I did then I gave her a kiss and told my Grandpa goodbye. My Grandpa had influenced me so much with his game that I knew decided to expand his game a little. Just click below and remember use what we just learned about the points of concurrency in a triangle to see if you can see which point of trajectory the ghost point is following.