Draw an initial circle, and arrange six circles tangent to it such that they touch both the original circle and their two neighbors. Then the three lines joining opposite points of tangency are concurrent in a point. The figures above show several possible configurations (Evelyn et al. 1974, pp. 31-37).
Letting the radii of three of the circles approach infinity turns three of the circles into the straight sides of a triangle and the central circle into the triangle's incircle. As illustrated above, the three lines connecting opposite points of tangency (with those along the triangle edges corresponding to the vertices of the contact triangle) concur (Evelyn et al. 1974, pp. 39 and 42).