159. Theorems of problems 118a) and b) transform into each other under polar transformation (therefore, it would be sufficient to prove only one of these theorems).

Theorems of problems 122a) and b) also transform into each other (and again, it would be sufficient to prove only one of these theorems).

The direct and inverse theorems of problem 123 transform into each other; and here the application of polar transformation does not lead to new results, because these two theorems are equivalent (see the remark at the end of the solution to problem 123).

Similarly, the application of polar transformation to the theorems of problems 125 and 126 does not lead to new results—these theorems also transform into themselves (only in new formulations of these theorems, obtained from those given in § 2 by the principle of duality, the perspective of triangles will be understood not as in the original theorems, but in the sense that the points of intersection of corresponding sides of the triangles lie on one line; by Desargues' theorem, these two definitions of perspective are equivalent).

The theorem of problem 127 transforms into the following: if three triangles $ABC, A_1B_1C_1$, and $A_2B_2C_2$ are arranged on a plane such that points $A, A_1, A_2$ lie on one line $q$, points $B, B_1, B_2$ lie on line $r$, and points $C, C_1, C_2$ lie on line $p$, and the three lines $p, q \parallel r$ pass through one point (see Fig. 325),

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Fig. 325.

then the three lines, on which, according to the theorem of problem 123, the points of intersection of corresponding sides of triangles $ABC$ and $A_1B_1C_1$, $ABC$ and $A_2B_2C_2$, $A_1B_1C_1$ and $A_2B_2C_2$ lie, intersect at one point (in other words: if the centers of perspective of three pairwise perspective triangles coincide, then their axes of perspective intersect at one point).

Theorems of problems 129a) and b) transform into each other as a result of polar transformation; since these two theorems are equivalent (see the solution to problem 129b)), it can also be said that these theorems transform into themselves.