The Fourth Side
Sweet triangle problem! Originally from B. F. Sherman's The Fourth Side of a Triangle.
What information do we need to know to determine a triangle is as old as geometry questions get. If we know the incircle, circumcircle and 9 point circle, is the triangle determined? Vsauce made a YouTube short which is where I saw the problem.
A. P. Guinand showed that you cannot make a Euclidean construction of a triangle from that information.
All the sides of a triangle satisfy:
a) Each side has its endpoints on the circumcircle;
b) each side has its midpoint on the nine-point circle; and
c) each side touches the incircle.
It turns out there is a fourth segment that satisfies these conditions!
Sherman used some sweet calculations to come up with two methods to find this fourth side.
Method 1: For a point M on the nine point circle, make the chord perpendicular to the orthocenter through M to the circumcircle. Find the point T on that line perpendicular to the incenter I. M, T, and O are 3 corners of a rectangle. The fourth corner P, when it is the inradius distance away from M (on the dotted red circle in this applet) indicates that the segment meets the conditions.
Method 2: For a point U on the circle with diameter spanning the incenter and the orthocenter, construct a segment through the orthocenter, and mark points on that segment the inradius away from U. When either of those points is on the 9 point circle, that's a segent that satisfies the conditions. This will happen 4 times. The three sides, and the fourth side!
Wow!