DesarguesInvolution3

Special case of Desargues involution theorem

Actually this is a special case of the special case discussed in DesarguesInvolution2.html . There we saw that a line (e) intersects the conics passing through {A,B,C} and tangent to a given line (t) at A, in pairs of points in involution.

Letting C coincide with B we obtain the figure below, in which Z₁, Z₂ coincide with Z.

Desargues theorem now asserts that Z is a double point of the involution defined on line (e) by the intersection points W₁, W₂ of conics passing through {A,B} and being there tangent to two given lines.
Note that the other fixed point Z' of the involution on (e) is the harmonic conjugate of Z with respect to W₁, W₂ (or X₁, X₂). This can be seen by considering the circle bundle generated by the two circles with diameters W₁W₂ and X₁X₂. The reason for this is explained in InvolutionBasic2.html .
The above property can be used to construct the conic passing through three points and two tangents. This is discussed in Conic3Pts2Tangents.html

References

Doerrie, Heinrich 100 Great Problems of Elementary Mathematics New York, Dover, 1965, p. 269.

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Special case of Desargues involution theorem

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References