Write-Up #4: Centers of a Triangle
Part A: Investigations
The centroid(G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.
The orthocenter (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side. (Note: H may not always be in the triangle.)
The circumcenter (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle and is the center of the circumcircle (the circumscribed circle) of the triangle. (Note: C many be outside of the triangle.)
The incenter (I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then the Incenter (I) lies on the angle bisector of each angle of the triangle. The incircle (inscribed circle) must use the shortest distance to the sides; hence the radius of the incircle is along the orthogonal line.
The medial triangle is a triangle that connects the three midpoints of the sides. It is similar to the original triangle and is 1/4 of its area.
The orthic triangle is an acute triangle that connects the feet of the altitudes.
The Nine-Point Circle is a circle that passes through the three mid-points of the sides, the three feet of the latitudes, and the three mid-points of the segments from the respective vertices to orthocenter, and has center at N, which corresponds to the Circumcenter, C.
Part B: Investigation of the Orthocenter of a Triangle
The orthocenter of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side and may be external to the triangle.
Given a triangle, construct one altitude from the top vertex of a triangle as shown.
In a similar manner construct the other three orthogonal projections from the other vertices:
Now if we hide the construction lines (ie, the altitude lines), then we can observe the orthocenter H, and the effects of different situations with an obtuse triangle. From the above example, the orthocenter is inside the triangle since the projections are inside the triangle.
If the triangle is obtuse (ie if the left vertex is an obtuse angle - an angle between 90 and 180 degrees), then the orthocenter is observed to be outside of the triangle because the (orthogonal) projection falls outside the triangle as shown below.
If we observe the loci of the orthocenter while the upper vertex traces a line from left to right, then H traces an arc as follows:
Hence, the orthocenter falls inside the triangle for acute angles and outside the triangle for obtuse angles.
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