MathDB

5

Part of 2012 239 Open Mathematical Olympiad

Problems(2)

Cyclic trapezoid

Source: 239 2012 S5

7/30/2020
Point MM is the midpoint of the base ADAD of trapezoid ABCDABCD inscribed in circle SS. Rays ABAB and DCDC intersect at point PP, and ray BMBM intersects SS at point KK. The circumscribed circle of triangle PBKPBK intersects line BCBC at point LL. Prove that ∠LDP=90∘\angle{LDP} = 90^{\circ}.
geometrytrapezoid
Tangent line to the circumcircle

Source: 239 2012 J5

7/30/2020
On the hypotenuse ABAB of the right-angled triangle ABCABC, a point KK is chosen such that BK=BCBK = BC. Let PP be a point on the perpendicular line from point KK to the line CKCK, equidistant from the points KK and BB. Also let LL denote the midpoint of the segment CKCK. Prove that line APAP is tangent to the circumcircle of the triangle BLPBLP.
geometrycircumcircle