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Trigonometry/Circles and Triangles/The Seven circles theorem

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The seven circles theorem states that if you take a circle, and arrange six circles around it so that they all touch it and each of the six touches its two neighbours, then the three lines joining opposite pairs of contact points are concurrent.

NB: this works even if six of the circles are inside the seventh.


Take any triangle and draw its incircle. Draw the three circles that touch the incircle and two of the other sides of the triangle. The lines joining the points where each of these tree circles touches the incircle to the point where the opposite side touches the incircle are concurrent.

This follows from the theorem by allowing three circles to have infinite radius.