MathDB

5

Part of 2016 Oral Moscow Geometry Olympiad

Problems(2)

excenters and projections lead to concurrent lines

Source: 2016 Oral Moscow Geometry Olympiad grades 8-9 p5

7/30/2019
Points IA,IB,ICI_A, I_B, I_C are the centers of the excircles of ABCABC related to sides BC,ACBC, AC and ABAB respectively. Perpendicular from IAI_A to ACAC intersects the perpendicular from IBI_B to BCB_C at point XCX_C. The points XAX_A and XBX_B. Prove that the lines IAXA,IBXBI_AX_A, I_BX_B and ICXCI_CX_C intersect at the same point.
geometryconcurrencyexcenterexcentersconcurrent
fixed point for a circle, tangent to a given line

Source: 2016 Oral Moscow Geometry Olympiad grades 10-11 p5

7/29/2019
From point AA to circle ω\omega tangent ADAD and arbitrary a secant intersecting a circle at points BB and CC (B lies between points AA and CC). Prove that the circle passing through points CC and DD and touching the straight line BDBD, passes through a fixed point (other than DD).
geometryFixed pointfixedtangent