where H

Two such triangles are called harmonic perspective.

Every conic passing through the vertices A,A',C',C has the pair (O,e) as a pair of pol-polar. Thus, if B lies on such a conic, then necessarily also B' will lie on c.

This implies that:

Through the vertices of every pair of harmonic perspective triangles passes a conic, which is invariant under the perspectivity defined by the two triangles.

By the general properties of perspectivities (see Perspectivity.html ), the two perspective triangles define a perspectivity. By the same general properties A'C and AC' intersect on e and corresponding property holds also for the pairs of line (BC', B'C) and (A'B, AB'). It follows that e is the polar of O, thereby proving the claim.

k = (A,A',O,A*) = (B,B',O,B*) = (C,C',O,C*),

defined along lines OA, OB, OC by their corresponding intersections A*, B*, C* with the perspectivity axis.

There is a conic passing through all six vertices of the triangles precisely when this k = -1.

A special case of perspective triangles is examined in the file Harmonic_Perspective_Triangles.html .

Harmonic_Perspective_Triangles.html

Desargues.html

Desargues_Many.html

Harmonic_Perspectivity.html

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