## Saturday, February 06, 2016

### [mspjzsai] Art of triangle centers

A diagram of some triangle centers could make for good art: look at it and think about it for a long time.  (Previously similar.)

We need a method of annotating relationships between points, lines, angles, areas, and circles.  Some of the relationships will be by construction (so less interesting), and some will be theorems.  Easiest is text annotation off to the side.  Color might be helpful.

Multiple copies of the diagram with triangles of different shapes, e.g., acute and obtuse.

Classic: Euler Line and Nine-Point Circle

Consider a diagram of only with the triangle, marked points, and the final Nine-Point Circle and Euler Line, in order to decrease the busyness of the diagram.  Extensions of the triangle sides needed for altitudes.

Triangle A0B0C0

A0A1 perpendicular B0C0 by construction (bc), B0B1 perpendicular A0C0 bc, C0C1 perpendicular A0B0 bc. Theorem (thm) A0A1 B0B1 C0C1 intersect at orthocenter H.

A0B0 midpoint C2 bc, B0C0 midpoint A2 bc, C0A0 midpoint B2 bc.  Thm A0A2 B0B2 C0C2 intersect at centroid G.

A0 B0 C0 lie on a circle centered on the circumcenter O bc.

A0H midpoint A3 bc, B0H midpoint B3 bc, C0H midpoint C3 bc.

Thm A1 A2 A3 B1 B2 B3 C1 C2 C3 lie on the Nine Point Circle.  The center of the circle is F bc.

Thm H G O F lie on the Euler Line.