1) AD is the bisector of ∠BAC, and the point P lies on the rye AD, so ∠BAP congruent ∠CAP.
2) ∠CPD =∠BPD
∠CPD+∠CPA =180⁰, as supplementary angles,
∠BPD+∠BPA=180⁰, as supplementary angles, so
∠CPD+∠CPA = ∠BPD+∠BPA, and because ∠CPD =∠BPD,
we have ∠CPA=∠BPA.
3)AP is a common side.
4) So, we have
∠BAP = ∠CAP, AP is a common side, ∠CPA=∠BPA.
And we have ΔCAP =ΔBAP, by the theorem ASA (angle, side,angle).
5) Because ΔCAP =ΔBAP, all corresponding sides and angles of theses triangles are congruent by CPCTC ( corresponding parts of the congruent triangles are congruent). So, corresponding sides CP and BP are congruent, CP=BP.