Journey to the Center of a Triangle: Some Hints

Consider these possible "centers" for a triangle. Maybe they will help you to decide what the center of a triangle might be, and to decide which center is the correct center for each of the 3 questions asked in Journey to the Center of a Triangle:
 The perpendicular bisector of a segment is the set of points equidistant from the endpoint of the segment. In the figure on the left, we see that the perpendicular bisectors of the 3 sides of a triangle meet at a point that is equidistant from the 3 vertices of a triangle. Since the distance from C to each of the 3 vertices is equal, C is the center of a circle which passes through each of these 3 points. For this reason, this point is called the Circumcenter of the triangle. Interestingly, the circumcenter does not always fall inside the triangle! In the figure on the right, the triangle is an obtuse triangle, and the perpendicular bisectors of the sides meet at a point outside the triangle. Where do you think the circumcenter would fall in a right triangle? The answer is interesting, and relates to the properties of angles inscribed in circles!

 The medians of a triangle meet at a point around which the area is equally distributed. This point is called the Centroid. Since the area is equally distributed around this point, if the triangle is a uniform thickness then the weight is equally distributed around this point also. Therefore the centroid it is the center of balance, or the center of gravity of the triangle. The area of the yellow triangle is equal to the area of the green triangle because their bases are equal (due to the midpoint) and they share the same altitude (the red segment). This is true for all pairs of triangles all the way around the centroid.

 The angle bisectors of a triangle meet at a point inside the triangle. Since each angle bisector is equidistant from the 2 sides of the angle that it bisects, this point is equidistant from all 3 sides of the triangle. Since this point is equidistant from all 3 sides of the triangle, if we construct a perpendicular from the incenter to a side, and use this segment as radius, we can inscribe a circle in the triangle. That is why this point is called the Incenter.

 The 3 altitudes of a triangle meet at a point called the Orthocenter. Unfortunately, the orthocenter has no interesting properties, at least, no simple ones. The altitudes of a triangle, by definition, must go from a vertex perpendicular to the opposite side. Therefore, in an obtuse triangle, 2 of the 3 altitudes extend outside the triangle! And so, their point of intersection (the orthocenter) must fall outside of the triangle also. Where do you suppose the altitudes are for a right triangle? And where would the orthocenter fall?
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Forward to the Answers to the questions asked in Journey to the Center of a Triangle