HARMONIC CONJUGATES

 Interactive Applet

You can move the points B, C and X (click on the point and drag it). The point X' is the harmonic conjugate of the point X with respect to the segment BC.
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

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 Information

Given three collinear points B, C and X, with B different to C, we say X' is the harmonic conjugate of X with respecto to the segment BC when

BX/CX = BX'/CX'.

This definition uses the notion of distance which is not a term defined in projective geometry. Unexpectedly, it's possible to define it using only intersections. More precisely, to construct the harmonic conjugate of the point X with respect to the segment BC:

1. Take any point A outside the line BC and construct the lines AB and AC.
2. Mark an arbitrary point P on the line AX and construct the lines BP and CP to intersect respectively the lines CA and AB at Y and Z.
3. Construct the line YZ to intersect BC at X'
4. X´ is the harmonic conjugate of X with respecto the segment BC.
Note that the middle point of the segment BC has no harmonic conjugate.

References:
Paul Yiu, An Introduction to the Geometry of the Triangle, 2001.

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.