![]() Prove that the feet of the perpendiculars from a point on the circumcircle to the sides (perhaps extended) are collinear. Prove that the perpendicular bisectors to the sides of the triangle are concurrent. Some ideas for write-ups about the Circumcenter (the circumscribed circle) of the triangle. Note: C may be outside of the triangle.Ĭ and explore its location for various shapes of triangles. The two points, C is on the perpendicular bisector of each side Lies on the perpendicular bisector of the segment determined by Since a point equidistant from two points Triangle is the point in the plane equidistant from the three Discuss the locus of the orthocenter as one side of a triangle is fixed and the opposite vertex is moved along a parallel line. Show how the proof of the concurrency of these perpendiculars follows from the proof of a circumcenter as the point of concurrency of the perpendicular bisectors of the side of a triangle. It the above figure on the right, the orthocenter H is outside the triangle (blue) and the three altitude segments are shown - none of them contain H. It is NOT NECESSARILY a point of concurrency of the altitudes. The Orthocenter is the point of concurrency of these three lines perpendicular to the lines along the opposite sides. Demonstrate and prove that the perpendiculars from a vertex to the line of the opposite side are concurrent. Some ideas for write-ups about the Orthocenter Sure your construction holds for obtuse triangles.) H and explore its location for various shapes of triangles. In some geometry texts the perpendicular LINE is the altitude rather than the segment from the vertex to the foot of the altitude. Rather, H lies on the lines extended along the altitudes. May be on the extension of the side of the triangle.) It shouldīe clear that H does not have to be on the segments that are theĪltitudes. An altitude is a perpendicular segment from a vertex Triangle is the point of concurrency or the common intersection of the three lines containing Use the centroid of a triangle to develop and prove a procedure for trisecting a line segment. In the quadrilateral there are two different centers of mass. Extend the concept of centroid to quadrilaterals. Show that the concurrency of the medians is a special case of Ceva's Theorem. Show that these triangles all have the same area. The medians divide the triangle into six small triangles. What does this mean? How can this idea be developed? The centroid is sometimes portrayed as the "center of mass" or "center of gravity" of the triangle. Prove the three medians of a triangle are concurrent and the centroid is the distance from a vertex to the midpoint of the opposite side. Some ideas for write-ups about Centroids: Use Geometer's Sketchpad (GSP) to ConstructĪnd explore its location for various shapes of triangles. Segment from a vertex to the midpoint of the opposite side. The CENTROID (G) of a triangle is the common Again the write-up is about the mathematical ideas, not just the pictures.ġ. medians) and explore some of the standard geometry for Help become familiar with GSP and to review some basic triangle These explorations are a set of activities to EMAT 6680 Explorations 04 - Centers of a Triangle
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