Geometry / Mr. Hansen 11/17/2003 [rev. 11/24/2004]

Name: ________________________

Mautner’s Lemma
Important note to anyone who may have reached this page on a Google search: When I published this
page, I was unaware that there was already a lemma in topology known as Mautner’s Lemma. The
problem on this page is completely unrelated and is merely named for a student of mine during the
2003-04 academic year at St. Albans. Sorry for the possible confusion.

Given: M and N are midpoints of two sides of DABC, as shown.
Prove: BN = CM Û AB = AC

In words: “Two medians of a triangle are congruent iff the triangle is isosceles.” Since this is a bidirectional result, you need to prove both the “if” direction and the “only if” direction. Starting with the triangle being isosceles is the easier direction.