Problem
GEO-B2-M04-P007 Centre of Homothety of Circles
#7
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A homothety with centre \(H\) sends a circle with centre \(O_1\) to a circle with centre \(O_2\). Prove that points \(H,O_1,O_2\) are collinear.
Where does a homothety send the centre of a circle-
A homothety sends a circle to a circle and preserves the central symmetry of the figure. Therefore the centre of the first circle is sent to the centre of the second circle: \(O_1 \mapsto O_2\). Under a homothety, a point, its image, and the centre of homothety are collinear. Hence \(H,O_1,O_2\) are collinear.
This is a useful lemma for problems with common tangents.