Euclidea 7.8

Euclidea 7.8 Circle Tangent to Three Lines
Construct a circle that passes through the midpoints of sides of the given acute triangle $$\triangle{\rm ABC} $$.



This problem uses a nice little Lemma that I actually didn't know before. The title of the problem is very suggestive: nine-point circle. Which nine points? It turns out that if we call the midpoints of the three sides as $$ {\rm D, E, F} $$, and the projection points of each of the vertices $$ {\rm A, B, C} $$ to the opposite side as $$ {\rm A', B', C'} $$, then all these six points, $$ {\rm D, E, F, A', B', C'} $$ are on the same circle!

To see why this is true, we prove only one of these points, $$ {\rm C'} $$, which we call $$ {\rm  G} $$ in this figure, lies on the circle passes through $$ {\rm D, E, F} $$.