The diameter of the circle O of BD, OA perpendicular to ob, M is a point on the inferior arc AB, the tangent MP of the circle O crosses through point m, the extension line of OA intersects at point P, and the intersection of MD and OA intersects at point n 1. Verify PM = PN. 2. If BD = 4, PA = two-thirds Ao, make BC ∥ MP through point B, intersect circle O at point C, and find the length of BC.

The diameter of the circle O of BD, OA perpendicular to ob, M is a point on the inferior arc AB, the tangent MP of the circle O crosses through point m, the extension line of OA intersects at point P, and the intersection of MD and OA intersects at point n 1. Verify PM = PN. 2. If BD = 4, PA = two-thirds Ao, make BC ∥ MP through point B, intersect circle O at point C, and find the length of BC.

1, ∵ PM is tangent ∵ PMO = 90 °= ∵ PMN + ∵ DMO ∵ Ao ⊥ Bo ∵ ODM + ∵ ond = 90 °∵ om = OD ∵ OMD = ∵ ODM ∵ PNM = ∵ ond ∵ PMD = ∵ PNM ∵ PM = pN2. In right triangle OPM, OM = OA ∵ BC ∥ MP ∥ OM