Problem
GEO-B2-M04-P018 Internal Centre of Two Circles
#18
★★★★☆ Level 4 of 5
Two circles have centres \(O_1\) and \(O_2\), radii \(6\) and \(9\), and \(O_1O_2=20\). Find the distance from \(O_1\) to the internal centre of homothety of the two circles.
The internal centre lies between the circle centres and divides \(O_1O_2\) in the ratio of the radii.
Let \(H\) be the internal centre of homothety. Then \(H\) lies on segment \(O_1O_2\) and \(\frac{HO_1}{HO_2}=\frac{6}{9}=\frac{2}{3}\). Let \(HO_1=x\), so \(HO_2=20-x\). We get \(\frac{x}{20-x}=\frac{2}{3}\). Hence \(3x=40-2x\), so \(5x=40\), \(x=8\). Therefore \(HO_1=8\).
Compare this with the external-centre problem so students do not confuse the two cases.