Is X a Midpoint of a Chord

29 December 2015, Created with GeoGebra

The applet above helps answer an inquiry by James Tanton on twitter.com and on his facebook page:

$X$ is a point inside a triangle. Is $X\;$ the midpoint of some line segment $AB$ with $A\;$ and $B\;$ on the triangle? If so, how find $A\;$ and $B?\;$ How many $AB\;$ pairs are there?

To get the answer, make a $180^{\circ}\;$ rotation around point $X\;$ - this is also called a half-turn. Let the original triangle be $ABC;$ and its image under the rotation $A'B'C'.\;$ The two necessarily intersect if $x\;$ lies in the interior of $\Delta ABC.$ The intersections come in pairs: there may be $1,$ $2,$ or $3\;$ intersections. Each pair is the segment that has $X\;$ as its midpoint. If $P,$ $Q,$ $R$ are the midpoints of $BC,\;$ $AC,\;$ and $AB,\$ respectively, then

  1. If $X$ lies in the interior of any of the triangles $AQR,\;$ $BPR,\;$ or $CPQ,\;$ then there is a unique chord joining points on the perimeter of $\Delta ABC$ with $X$ as its midpoint.
  2. If $X$ lies on the perimeter of $\Delta PQR,\;$ there are two such points.
  3. If $X$ lies on in the interior of $\Delta PQR,\;$ there are three such points.

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