Given triangle XYZ and a point P in its interior, drop perpendiculars from P to the side of the triangle at points A, B, and C as shown.
Prove: (XA)² + (YB)² + (ZC)² = (XC)² + (YA)² + (ZB)²

You can construct line segment PY but its not an angle bisector of angle Y. It does make two right triangles however.
Triangle Proofs

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    Consider the line segment XP.
    From Pythagoras:
    AX² + PA² = XP²
    CX² + PC² = XP²
    I: AX² + PA² = CX² + PC²...
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