in Nine Point Circle

THE
NINE - POINT
CIRCLE
RELATED
KNOWLEDGE
Related knowledge:
Perpendicular feet
 The point on the leg

opposite a given vertex of
a triangle at which the
perpendicular passing
through that vertex
intersects the side.
Related knowledge:
Orthocenter
 The intersection of
the three
perpendicular
feet(or altitudes) of
a triangle.
Related knowledge
Centroid (of a
triangle)
 The intersecting point of

three medians of a triangle.
 Median: A line segment
drawn from one vertex to
the midpoint of the
opposite side.
History of Nine-
Point Circle
1804 1821
Bevan's theorem appears in The nine points are explicitly
Leybourn's Mathematical Repository mentioned in Gergonne's Annales de
Mathematiques , volume xi., in an article by
1807 Brianchon and Poncelet. This article contains
John Butterworth poses a question the theorem establishing the characteristic
relating to nine-point circle in Gentleman's property of the nine point circle.
Mathematical Companion 1822
1808 First enunciation of Feuerbach's
Two solutions to Butterworth's Theorem, including the first published proof,
original question are given in appears in Karl Wilhelm
Feuerbach's Eigenschaften einiger
the Gentleman's Mathematical Companion,
merkwiirdigen Punkte des geradlinigen
one given by Butterworth himself, and the
Dreiecks, along with many other interesting
other by John Whitley proofs relating to the nine point circle.
Karl Wilhelm
Feuerbach was a
German Geometer who
discovered the nine
point circle of a triangle
.Including some
important properties of
nine-point circle.
BORN ON 30 May 1800 DIED 12 March 1834

What Is Nine –
Point Circle?
The nine-point circle (also known as Euler's
circle or Feuerbach's circle) of a given triangle is a circle
which passes through 9 "significant" points:
"The nine-point circle is tangent to the incircle, has a
radius equal to half the circumradius, and its center is
the midpoint of the segment connecting the orthocenter
and the circumcenter." -hankinjg
The circle has 9 points on the
circumference, which are:
 Three mid-points of the sides of a

triangle(M);
 Three perpendicular feet drawn

from the vertices of the triangle to
the nearest base(H); and
 The mid-points of the segments

that join the vertices and the
orthocenter(E).
The center of the nine-point circle is the
nine-point center and is usually denoted.
The nine-point circle is tangent to the incircle,
has a radius equal to half the circumradius,
and its center is the midpoint of the segment
connecting the orthocenter and the
circumcenter, upon which the centroid also
falls.
Properties of the
Nine-Point Circle
The Radius
The radius of the nine-
point circle is half of the radius
of its circumradius.
Circumcircle: For any triangle
there is always a circle passing
through its three vertices.
Circumradius: Radius of
circumcircle
The Center
The center of the nine-

point circle is the midpoint of
the Euler line.
Euler line: A line
which is produced by joining
the orthocenter and the
centroid of a triangle.
Feuerbach’s
Theorem
Feuerbach's theorem:
the nine-point circle
is tangent to the incircle
and
excircles of a triangle. The
incircle tangency is the
Feuerbach point.
The Feuerbach’s
Theorem
Incircle: The unique circle that
is tangent to each of the triangle's
three sides.
Excircle: The circle tangent to
the extended two non-adjacent sides
of a triangles and to the other side of
the triangle.
The Feuerbach’s Theorem
(Continue)
The nine-point
circle of a triangle
“touches” the incircle
and the three excircles.
Construction of
Nine – Point
Circle
Steps on how To
construct the nine
point circle of a
triangle
1. Draw a triangle ABC.
2. Construct the midpoints of
the three sides. Label them as
L, M, N.
3. Construct the feet of the
altitudes of the triangle ABC.
Label them as D, E, F. Label
the point of intersection of
the three altitudes as H. This
is also called
the orthocenter.
4. Construct the midpoints of
the segments AH, BH, CH.
Label them as X, Y, Z.
This circle is called
the Nine-Point
Circle.
To find the center of the
Nine-Point Circle,
construct the
circumscribed circle for
triangle LMN. Label the
center as U.
The center of the
circumscribed circle for
triangle LMN will also
be the center of the Nine-
Point Circle labeled as
U.
A second way to find
the Nine-Point Center
is to begin by
constructing the
circumcircle for
triangle ABC. Label
the circumcenter as
CC.
The center of the Nine-Point
Circle, U, is the midpoint from No matter what type of
the orthocenter, H, and the triangle we have, other
circumcenter, CC, of triangle than a degenerate
ABC. triangle, those nine
points will always lie in
a circle, the nine point
circle, with center at U.
THE END!
THANKS FOR
LISTENING!

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THE

 The point on the leg

 The intersecting point of

BORN ON 30 May 1800 DIED 12 March 1834

 Three mid-points of the sides of a

 Three perpendicular feet drawn

 The mid-points of the segments

The center of the nine-

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