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Having your magenta triangle ABC, you then incorporate a new vertex X. I think it is obvious that there will be two lines starting at D that will not intersect between any of the edges of the triangle ABC.

These two lines could be AX & BX, BX & CX or AX & CX. You can then treat your problem as the classical problem of "do two lines intersect"? You can then check which of this pairs of lines does not intersect with any of the ABC triangle edges following, for example, any of the methods from this questionany of the methods from this question. Hence, you will have the two new edges of the new triangle.

Having your magenta triangle ABC, you then incorporate a new vertex X. I think it is obvious that there will be two lines starting at D that will not intersect between any of the edges of the triangle ABC.

These two lines could be AX & BX, BX & CX or AX & CX. You can then treat your problem as the classical problem of "do two lines intersect"? You can then check which of this pairs of lines does not intersect with any of the ABC triangle edges following, for example, any of the methods from this question. Hence, you will have the two new edges of the new triangle.

Having your magenta triangle ABC, you then incorporate a new vertex X. I think it is obvious that there will be two lines starting at D that will not intersect between any of the edges of the triangle ABC.

These two lines could be AX & BX, BX & CX or AX & CX. You can then treat your problem as the classical problem of "do two lines intersect"? You can then check which of this pairs of lines does not intersect with any of the ABC triangle edges following, for example, any of the methods from this question. Hence, you will have the two new edges of the new triangle.

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Dan
Dan
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Having your magenta triangle ABC, you then incorporate a new vertex X. I think it is obvious that there will be two lines starting at D that will not intersect between any of the edges of the triangle ABC.

These two lines could be AX & BX, BX & CX or AX & CX. You can then treat your problem as the classical problem of "do two lines intersect"? You can then check which of this pairs of lines does not intersect with any of the ABC triangle edges following, for example, any of the methods from this question. Hence, you will have the two new edges of the new triangle.