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Home / Geometry / In the triangle ABC, AB>AC and the bisectors of the ∠B and ∠C intersect at the point P. Prove that PB>PC.

In the triangle ABC, AB>AC and the bisectors of the ∠B and ∠C intersect at the point P. Prove that PB>PC.

General enunciation: In the triangle ABC, AB>AC and the bisectors of the ∠B and ∠C intersect at the point P. Prove that PB>PC.

triangle

Particular enunciation: In the triangle ABC, AB>AC and the bisectors of the ∠B and ∠C intersect at the point P. It is required prove that PB>PC.
Proof: PB is the bisector of ∠B.
∴∠PBC=½∠B
PC is the bisector of ∠C.
We know,
In a triangle the angle opposite the greater side is greater than the angle opposite the smaller side since AB>AC
∴ ∠ACB>∠ABC
⟹½∠ACB>½∠ABC
⟹∠PCB>∠PBC
⟹PB>PC
Therefore PB>PC.(Proved)

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