MathDB

1.4

Part of Geometry Mathley 2011-12

Problems(1)

Geometry Mathley 1.4 concurrency of 4 lines

Source:

6/6/2020
Given are three circles (O1),(O2),(O3)(O_1), (O_2), (O_3), pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let (X1)(X_1) be the circle externally tangent to (O1)(O_1) and internally tangent to the circles (O2),(O3),(O_2), (O_3), circles (X2),(X3)(X_2), (X_3) are defined in the same manner. Let (Y1)(Y_1) be the circle internally tangent to (O1)(O_1) and externally tangent to the circles (O2),(O3)(O_2), (O_3), the circles (Y2),(Y3)(Y_2), (Y_3) are defined in the same way. Let (Z1),(Z2)(Z_1), (Z_2) be two circles internally tangent to all three circles (O1),(O2),(O3)(O_1), (O_2), (O_3). Prove that the four lines X1Y1,X2Y2,X3Y3,Z1Z2X_1Y_1, X_2Y_2, X_3Y_3, Z_1Z_2 are concurrent.
Nguyễn Văn Linh
concurrentgeometrycirclestangent circles