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User:Ghazer~enwiki/bisector

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Generalization

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Generalized angle bisector says that if D lies on the line BC, and the point A doesn't, the following is true:

Proof of generalization

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If we define B1 and C1 as the bases of altitudes in the triangles ABD and ACD through, respectively, B i C, it is true that:

It is also true that both the angles DB1B and DC1C are right, while the angles B1DB and C1DC are congruent if D lies on the segment BC and they are identical otherwise, so the triangles DB1B and DC1C are similar (AAA), which implies:

Q.E.D.