9. Construct triangle ABC, its incircle, its three excircles, and its nine-point circle. Conjecture? Proof?
The incircle is the circle that is tangent to triangle whose center is the point of concurrency of the angle bisectors of that triangle.
The excircle of a triangle is the circle whose center is the intersection of the external angle bisectors of two adjacent angles. It is tangent to the side of the triangle that is between the two angles. The excircle is also tangent to the extension of the other two sides of the triangle. Every triangle has 3 excircles.
The 9 point circle contains the midpoint of each side of a triangle, the foot of each altitude, and the midpoint of the segment of each altitude from its vertex to the orthocenter (point where the altitudes meet). The 9 point circle’s center is the midpoint of the segment that connects the orthocenter and the circumcenter of a triangle.
The picture below shows the combined construction of the incircle, the three excircles, and the 9 point circle.
The red circles are the excircles, the green circle is the incircle, and the purple circle is the 9 point circle.
Notice that the 9 point circle is tangent to the other 4 circles. The excircles are all on the outside of the 9 point circle while the incircle will always be on the interior of the 9 point circle. This conjecture is known as Feuerbach’s Theorem.
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