MathDB

3

Part of 2010 Oral Moscow Geometry Olympiad

Problems(2)

collinear wanted, intersecting circles and tangents related

Source: 2010 Oral Moscow Geom3try Olympiad grades 8-9 p3

8/16/2020
Two circles w1w_1 and w2w_2 intersect at points AA and BB. Tangents ℓ1\ell_1 and ℓ2\ell_2 respectively are drawn to them through point AA. The perpendiculars dropped from point BB to ℓ2\ell_2 and ℓ1\ell_1 intersects the circles w1w_1 and w2w_2, respectively, at points KK and NN. Prove that points K,AK, A and NN lie on one straight line.
geometrycirclescollinear
fixed point, S_{KMC} + S_{KAC}=S_{ABC}, area condition , two non constant points

Source: 2010 Oral Moscow Geometry Olympiad grades 10-11 p3

5/8/2020
On the sides ABAB and BCBC of triangle ABCABC, points MM and KK are taken, respectively, so that SKMC+SKAC=SABCS_{KMC} + S_{KAC}=S_{ABC}. Prove that all such lines MKMK pass through one point.
geometryfixedFixed pointarea of a triangleareas