Triangle and Circles

JISOOK OH

 

 

The nine-point circle of a triangle is tangent to incircle and to the three excircle.

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The nine-point circle emerged in europe in the early 1800s. The French mathematician Jean-Victor Poncelet (1788-1867) named it the "Nine-Point Circle," and Karl W. Feuerbach (1800-1834) in Germany (where it is called the Feuerbach circle) made several remarkable discoveries concerning it. These properties are so unique. I'll mention about some properties.

 

PROPERTIES OF NINE-POINT CIRCLE AND NINE-POINT CENTER

1. The nine-point circle of triangle ABC with orthocenter H passes through th e midpoint L,M and N of the three sides, the feet of the altitudes D,E and F to those sides, and the Euler points x,y, and Z, which are the midpoints of the segments AH,BH, and CH, respectively.

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2. The nine-point circle center U lies on the Euler line of the triangle ABC.

 

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3. The tangents to the nine-point circle at the midpoints L,M, and N of the sides of the triangle form a triangle that is similar to the orthic triangle. In fact, the sides of this triangle are parallel to those of th eorthic triangle.

 

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4. Feuerbach's Theorem The nine-point circle is tangent to the incircle. Futhermore, the other three circles tangent to the extended sides of a triangle (called the excircles) are each tangent to the nine-point circle.


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