Terminology:
Circumcenter: (C) is the point formed from the intersection of the three perpendicular bisectors of the triangle. The is also one of the end points of the Euler Line Segment.
Orthocenter (H) is the point formed from the intersection of the three altitudes of the triangle. This is the other end point of the Euler Line Segment.
Acute Angle - an angle that has a value greater than 0 but less than 90 degrees.
Obtuse Angle - an angle that has a value greater than 90 but less than 180 degrees
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| (2) H is concurrent with Vertex A | ||
The characteristic property of a triangle with an orthocenter (H) that is
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| (2) C is concurrent with midpoint | ||
The characteristic property of a triangle with an circumcenter (C) that is
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To investigate the path of the orthocenter we will hold two of the vertexes (A & B) of the triangle constant while manipulating the other in a linear motion. This way will see the orthocenter make its transition through the triangle as the angles of the triangle increase or decrease.
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The orthocenter follows a parabolic shapes as it makes the transition through the triangle. The orthocenter will alway exit the triangles through the obtuse angle, such as in diagram 3 where angle B = 117 degrees. The orthocenter travels through the vertex B and then extends itself outside te triangle along the parabolic path.
(COOL, CAN I SEE THAT
HAPPEN NOW!!!)
To investigate the path of the circumcenter we will hold two of the vertexes (A & B) of the triangle constant while manipulating the other. This way will see the circumcenter make its transition through the triangle as the angles increase or decrease.
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In contrast to the orthocenter, the circumcenter travels a linear path. A path that is exactly on one of the perpendicular bisectors of the triangle. The circumcenter will always exit the triangle through the midpoint that is opposite the obtuse angle. The diagram to the left shows how the circumcenter has exited the triangle through the midpoint opposite the obtuse angle A.
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Extending our previous investigation on the orthocenter we now look at manipulating all 3 verticies of the triangle. We will move the vertex E in a east/west linear direction while moving the side AB in a north/south linear direction. The result is an envelope of parabolic paths. A number of interesting findings came out of this invetigation but I will leave it as a further investigation for the reader.
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Now if we look at the Euler line in terms of moving all 3 verticies at a time. We will move the vertex E in a east/west linear direction while moving the side AB in a north/south linear direction.
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Some of our earlier findings concerning the parabolic shape created by the path of orthcenter seem to be some what modified here. A cusp occurs diving the parabola into two symmetrical shapes. The cusp is formed from the linear path of the circumcenter as it extends outside the coundry of the orthocenter. The triangular shape that can occur is also a product of the circumcenters path. Further investigation - what is the relationship between the orthocenter and the circumcenter so that the cusp is no longer there.
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right triangle. The
circumcenter is concurrent with the midpoint (Diagram 2) because if the
central angle of a circle is 180 degrees, then the inscribed on that angle
is 90. This is the case that we have to the left.