# Construction of the Contraharmonic Mean in a Trapezoid

## by Shannon Umberger

Given trapezoid KLMN, with KL // NM, KL = a,
and NM = b.

Construct the arithmetic mean segment (in red)
and the harmonic mean segment (in green).

Construct the midpoint of the harmonic mean
segment and call it point R. Construct a line perpendicular to
the harmonic mean segment at point R. Construct the intersection
of this line and the arithmetic mean segment and call it point
S.

Construct a circle with center at point S and
passing through point R. Construct the intersection of this circle
with the above line and call it point T. Since SR and ST are radii
of the same circle, then SR = ST.

Construct a line parallel to base NM through
point T. Construct the intersection of this line and leg KN and
call it point P. Construct the intersection of the same line and
leg LM and call it point Q.

Construct segment PQ. The length of this segment
is the contraharmonic mean, "c," of the bases KL and
NM.

Double check the construction by taking measurements
and using the equation for the contraharmonic mean.

**Return
to Essay # 3 - Some "Mean" Trapezoids**