Notes on Antireciprocal Points

Extract

Definition. If x, y, z and ξ, η, ζ be the perpendiculars on the sides BC, CA, AB of the Δ ABC from points O and O′, then O and 0′ are antireciprocal points if xξ η, zζ:: tanA : tanA : tanB: tanC.

I. Construction to find a point antireciprocal to O (Fig. 4).

Draw through O a line MN antiparallel to BC. Draw OY perpendicular to AC, and OZ perpendicular to AB. Draw lines parallel to AB and AC, and at distances from them respectively equal to YN and MZ, and let them cut in P. Join AP. Find a similar line BQ, and let AP and BQ cut in O′.