Theorems on concurrence of lines, segments, or circles associated
with triangles all deal with three or more objects passing through
the same point. The theorems that follow are fundamental
proofs that should be part of secondary school geometry.
1. Prove: The perpendicular bisector
of the sides of a triangle meet at a point which is equally distant
from the vertices of the triangle.
2. This point of concurrence is the circumcenter
of the triangle and the circle through the three vertices with
center at the point of concurrence is the circumcircle.
Show a construction for the circumcircle of any triangle.
3 Prove. The bisectors of the angles
of a triangle meet at a point that is equally distant from the
sides of the triangle.
4. This point of concurrence is the incenter
of the triangle and the circle with center at the incenter and
tangent to each of the three sides of the triangle is the incircle.
Show a construction for the incircle of any triangle.
5. Prove. Let the sides of a triangle
be extended so as to indicate the external angles. The
bisectors of the external angles on one side of a triangle and
the bisector of the opposite internal angle are concurrent.
6. This point of concurrency of the two external
angle bisectors and the corresponding opposite internal angle
bisector is an excenter. The circle with center at the
excenter and tangent to the lines of the sides of the triangle
is an excircle. Each triangle has three excenters and three
excircles. Show a construction for the excircles of a triangle.
Prove. The medians of a triangle are concurrent and intersect
each other in a ratio of 2:1.
8. Prove. The perpendiculars from each
vertex to the line of the opposite side of a triangle are concurrent.
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