Viviani's theorem
Revision as of 20:42, 9 November 2012 by DaChickenInc (talk | contribs)
The Viviani's Theorem states that for an equilateral triangle, the sum of the altitudes from any point in the triangle is equal to the altitude from a vertex of the triangle to the other side.
Proof
Let be an equilateral triangle and
be a point inside the triangle.
We label the altitudes from
to each of sides
,
and
,
and
respectively. Since
is equilateral, we can say that
WLOG. Therefore,
,
and
. Since the area of a triangle is the product of its base and altitude, we also have
. However, the area of
can also be expressed as
. Therefore,
, so
, which means the sum of the altitudes from any point within the triangle is equal to the altitude from the vertex of a triangle.