Question 524163
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<i><b>Centroid:</b></i>


Calculate the coordinates of the mid-points of AC and BC. 


Use the mid-point formulas:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}] and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}]


Write an equation for a line that passes through point A and the mid-point of BC.


Use the two-point form of an equation of a line:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ \left(\frac{y_1\ -\ y_2}{x_1\ -\ x_2}\right)(x\ -\ x_1) ]


where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the given points.


Then write an equation for a line that passes through point B and the mid-point of AC


Using any convenient method, solve this 2X2 system.  The solution set will be the centroid of the triangle -- the point of intersection of the three triangle medians.


<i><b>Orthocenter:</b></i>


Calculate the slope of sides AB and BC of the triangle using the slope formula:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\ =\ \frac{y_1\ -\ y_2}{x_1\ -\ x_2} ]


where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the given points. 


Then, using the point-slope form of an equation, and the fact that perpendicular lines have slopes that are negative reciprocals, write equations of the two altitudes to sides AB and BC -- lines perpendicular to AC and passing through B and perpendicular to BC and passing through A.


Solve the 2X2 system.  The intersection of the two altitudes is the orthocenter.


<i><b>Circumcenter</b></i>


Using the slopes calculated above for AB and BC and the mid-points calculated for the Centroid solution, write equations of the perpendicular bisectors of AB and BC.  Perpendicular to AB and passing through the mid-point of AB, then perpendicular to BC and passing through the mid-point of BC.


Solve the 2X2 system.  The intersection of the perpendicular bisectors is the circumcenter (a point equidistant from the three vertices and therefore the center of a circle that passes through the three vertices of the triangle.)


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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