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triangle ABC and triangle DBC are two is...

`triangle` ABC and `triangle` DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that, ( i ) `triangle ABD = triangle ACD (ii) triangle ABP = triangle ACP` (iii) AP bisects `angle` A as well as `angle` D. (iv) AP is the perpendicular bisector of BC

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