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Your conjecture is a theorem in Russell, Pure Geometry (1893), pg 153 (Ch. XIV, Art. 3):

Two triangles which are reciprocal for a conic are homologous ; and conversely, if two triangles be homologous they are reciprocal for a conic.

In this book, "homologous" means "in perspective" and "reciprocal" means that the vertices and sidelines of the two triangles are in a pole/polar relationship of the triangle vertices and sidelines.

The problem of constructing of a conic from two perspective triangles is converted to that of constructing a conic from a self-conjugate (i.e. self-reciprocal) triangle and a pole-polar pair (described in Ch. XXV, Art 12, pg 234-235).

The latter construction can be done with ruler and compass, with the snag that some of the constructed points may be imaginary. Midway through, here's how Russell deals with this caveat:

This completes the theoretical solution of the problem ; and we have shown that one, and only one, conic can be drawn satisfying the given conditions. Practically the above solution is worthless ; for any pair of the points XX', YY', ZZ' may be imaginary. The following is the practical construction when the conic is real.

Your conjecture is a theorem in Russell, Pure Geometry (1893), pg 153 (Ch. XIV, Art. 3):

Two triangles which are reciprocal for a conic are homologous ; and conversely, if two triangles be homologous they are reciprocal for a conic.

In this book, "homologous" means "in perspective" and "reciprocal" means the pole/polar relationship of the triangle vertices and sidelines.

The problem of constructing of a conic from two perspective triangles is converted to that of constructing a conic from a self-conjugate (i.e. self-reciprocal) triangle and a pole-polar pair (described in Ch. XXV, Art 12, pg 234-235).

The latter construction can be done with ruler and compass, with the snag that some of the constructed points may be imaginary. Midway through, here's how Russell deals with this caveat:

This completes the theoretical solution of the problem ; and we have shown that one, and only one, conic can be drawn satisfying the given conditions. Practically the above solution is worthless ; for any pair of the points XX', YY', ZZ' may be imaginary. The following is the practical construction when the conic is real.

Your conjecture is a theorem in Russell, Pure Geometry (1893), pg 153 (Ch. XIV, Art. 3):

Two triangles which are reciprocal for a conic are homologous ; and conversely, if two triangles be homologous they are reciprocal for a conic.

In this book, "homologous" means "in perspective" and "reciprocal" means that the vertices and sidelines of the two triangles are in a pole/polar relationship.

The problem of constructing a conic from two perspective triangles is converted to that of constructing a conic from a self-conjugate (i.e. self-reciprocal) triangle and a pole-polar pair (described in Ch. XXV, Art 12, pg 234-235).

The latter construction can be done with ruler and compass, with the snag that some of the constructed points may be imaginary. Midway through, here's how Russell deals with this caveat:

This completes the theoretical solution of the problem ; and we have shown that one, and only one, conic can be drawn satisfying the given conditions. Practically the above solution is worthless ; for any pair of the points XX', YY', ZZ' may be imaginary. The following is the practical construction when the conic is real.

Source Link
brainjam
brainjam
  • 9.3k
  • 1
  • 23
  • 45

Your conjecture is a theorem in Russell, Pure Geometry (1893), pg 153 (Ch. XIV, Art. 3):

Two triangles which are reciprocal for a conic are homologous ; and conversely, if two triangles be homologous they are reciprocal for a conic.

In this book, "homologous" means "in perspective" and "reciprocal" means the pole/polar relationship of the triangle vertices and sidelines.

The problem of constructing of a conic from two perspective triangles is converted to that of constructing a conic from a self-conjugate (i.e. self-reciprocal) triangle and a pole-polar pair (described in Ch. XXV, Art 12, pg 234-235).

The latter construction can be done with ruler and compass, with the snag that some of the constructed points may be imaginary. Midway through, here's how Russell deals with this caveat:

This completes the theoretical solution of the problem ; and we have shown that one, and only one, conic can be drawn satisfying the given conditions. Practically the above solution is worthless ; for any pair of the points XX', YY', ZZ' may be imaginary. The following is the practical construction when the conic is real.