Question

use the construction of The circumcircle to construct Euler's line. Describle your construction in detail

Answer 1

1. Using a compass and straightedge, construct point of concurrency on individual given triangles. Leave all construction markings, do not erase any of your work.

1. Circumcenter – Construct all perpendicular bisectors in a triangle. Where they concurrently meet is the circumcenter. Label it as O. The circumcenter is equidistant from all three vertices of the triangle. Once you have found the circumcenter,use this center to draw the circle that will circumscribe about the triangle.
2. Incenter – Construct all angle bisectors in the triangle. Where they concurrently meet is the incenter. Label it as I. The in center in equidistant from the sides of the triangle. Once you have found the incenter, use this center to inscribe a circle inside this triangle.
3. Centroid – Construct all medians in the triangle. Where they concurrently meet is the centroid. Label it as G. The centroid is also known as the center of gravity for the triangle. The centroid is also known as the balance point of the triangle.
4. Orthocenter – Construct all altitudes in the triangle. Where they concurrently meet is the orthocenter. Label it as H.
5. In 1765, Leonhard Euler (pronounced “Oiler”) proved that, for any triangle, three of the four points of concurrency are collinear. The line that contains these three points is known as Euler’s Line. Which point of concurrency does not necessarily lie on Euler’s Line?
6. For ?ABC, construct Euler’s Line. Label the centroid as G, the circumcenter as O, and the orthocenter as H. Highlight Euler’s Line.

Construction of the Nine-Point Circle

The nine-point circle is also known as Euler’s Circle. It is named for the nine easily constructed points that lie on the circumference. This circle also has many exciting properties. Every triangle has a nine-point circle which is connected to both it’s inscribed circle and circumscribed circle, and of course Euler’s Line.

STEPS

1. Given ?ABC, construct the midpoints of the sides. Label them L, M, and, N.

1. Construct the feet of the altitudes of ?ABC. Label them D, E, and F. Label the point of the intersections of the three altitudes as H. This is the orthocenter.

1. Construct the midpoints of the segments AH, BH, and CH. Label them as X, Y, and Z.  
2. Notice that the points L, M, N, D, E, F, Y, Z lie in a circle. This is the nine-point circle!
3. To find the center of this circle, connect L, M, and N to form a triangle. Now find the circumcenter of ?LMN by constructing the perpendicular bisectors of this triangle. Label the circumcenter as U.
4. Point U is now the center of the circle that will pass through L, M, N, D, E, F, X, Y, Z. Use your compass to carefully construct the 9-point circle.
5. Connect H and U. This is once again Euler’s Line. Highlight Euler’s Line.