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AD^3 = AB x AC x AP, equal sngles

Source: 1999 Romania NMO IX p1

August 14, 2024

equal anglesgeometry

Problem Statement

Let

AD

be the bisector of angle

A

of the triangle

ABC

. One considers the points M, N on the half-lines

(AB

and

(AC

, respectively, such that

\angle MDA = \angle B

and

\angle NDA = \angle C

. Let

AD \cap MN=\{P\}

. Prove that:

AD^3 = AB \cdot AC\cdot AP

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