Diagonal In A Relation

Geometry Level 4

Let A B C D ABCD be a convex quadrilateral. The diagonals A C AC and B D BD intersect at K K . Show that A B C D ABCD is cyclic if and only if

A K sin ⁡ A + C K sin ⁡ C = B K sin ⁡ B + D K sin ⁡ D AK \sin A + CK \sin C = BK \sin B + DK \sin D B K sin ⁡ A + A K sin ⁡ B = D K sin ⁡ C + C K sin ⁡ D BK \sin A + AK \sin B = DK \sin C + CK \sin D A K sin ⁡ A + B K sin ⁡ B = C K sin ⁡ C + D K sin ⁡ D AK \sin A + BK \sin B = CK \sin C + DK \sin D C K sin ⁡ B + A K sin ⁡ D = D K sin ⁡ A + B K sin ⁡ C CK \sin B + AK \sin D = DK \sin A + BK \sin C

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Danish Ahmed by Danish Ahmed , 24, India

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