One of the topics typically covered in high school geometry is line segments associated with triangles. Specifically, these include medians, altitudes, angle bisectors and perpendicular bisectors. After teaching this subject off and on for about nine years, it occurred to me that although I often taught that for any given triangle, the medians are concurrent, the altitudes are concurrent, etc. , I could not recall a justification of these facts. The following is an attempt to verify some of these ideas.
We will begin with a discussion of Ceva's theorem and its converse which will be used in subsequent proofs. Click here for a proof of Ceva's theorem and its converse.
Theorem 1: The medians of a triangle are concurrent.
Theorem 2: The altitudes of a triangle are concurrent.
Theorem 3: The angle bisectors of a triangle are concurrent.
(Reference: Coxeter, H. S. M. & Greitzer, S. L. (1967). Geometry Revisited. Washington D. C.: The Mathematical Association of America. pp. 4-10.)
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