Euclidea 6.9

Euclidea 6.9 Nine Point Circle
Construct a circle that passes through the midpoints of sides of the given acute triangle $$\triangle{\rm ABC} $$.

This is a very interesting little problem, using symmetry and a special way to construct parallel lines that is different from those in chapter 5. We only present the 6E solution here.

Solution 6E



 * 1) Draw circle $$ \odot{\rm A} $$ and $$ \odot{\rm B} $$, with radius $$ \overline{\rm AB} $$, just like making a bisector of $$ \overline{\rm AB} $$, get two intersecting points  $$ \odot{\rm D, E} $$;
 * 2) Draw circle $$ \odot{\rm E} $$ with radius $$ \overline{\rm EC} $$;
 * 3) Draw circle $$ \odot{\rm D} $$ with radius $$ \overline{\rm DC} $$, intersect $$ \odot{\rm E} $$ at $$ {\rm F} $$, connect line $$ \overline{\rm CF} $$;