# The Cleaver: What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet may suggest the following statment:

In ΔABC, let CL be the angle bisector of angle C, M the midpoint of AB, and MD||CL. (D lies on the longest of AC and BC.) Then points M and D split the perimeter of ΔABC into equal halves.

Assume AC > BC. Extend AC beyond C to F such that,

AD = BC + CD.

By construction,

AM + BM,

so that

AM + AD = BM + BC + CD.

The line MD that joins a midpoint of a side with the opposite perimeter-bisecting point is called a *cleaver*. The three cleavers in a triangle intersect at the Spieker center of the triangle.

### References

- R. Honsberger,
*Episodes in Nineteenth and Twentieth Century Euclidean Geometry*, MAA, 1995, pp. 2-4

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Copyright © 1996-2018 Alexander Bogomolny