Natalie Smith

EMAT4680

Java Explorations

Centroid and midpoint observations.

When the vertices are changed to form obtuse, right and equilateral triangles, the ration of the areas of ABG and AGC stays equal to one. This is interesting because the angle bisector through the midpoint does not always create triangles that are the same shape. When point G is moved along the segment from B to C the ration increases until it becomes undefined when the area of triangle AGC is one and the ratio equation is undefined. When moved from point C to point B, the ration steadily decreases to zero when the area of triangle ABG is zero. Another observation is that the centroid seems to divide the angle bisector through the median into equivalent ratios no matter the triangle shape. The ratio looks to be 2 to 1 when comparing the distance from the centroid to the midpoint and then the centroid to the vertex. If point A moved to the same point as the midpoint of BC (point G), then the distance between the centroid and segment GA is zero. So when the triangle resembles more closely a line, then the centroid actually lies on the line.

Four centers of triangles

My first thought about having four centers of triangles is that it is a contradictory notion. I wondered how you could have four distinct centers of triangles. Obeservation and definition showed me how the centers are actually centers of other figures/lines with respect to the given triangle. This java script clearly illustrated that with an equilateral triangle, the four centers are at one point that seems to be equidistant from each of the triangle vertices. When the triangle is right, the points all rest in a line, but the orthocenter rest on the vertex of the right angle and the cicumcenter lies on the midpoint of the hypotentuese. That observation seems to reveal a little about trigonometry principles. When the triangle becomes obtuse, the only points that remain inside the triangle are the incenter and the circumcenter. When a vertex is shifted on top of another, the centroid seems to divide the resulting "segment" at the above-mentioned 2 to 1 ratio and the incenter moves to the remaining vertex.