Nine-point Circle - Tangent Circles

Tangent Circles

In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He postulated that:

... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle... (Feuerbach 1822)

The point at which the incircle and the nine-point circle touch is often referred to as the Feuerbach point.