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Confusing Circles

Note by Priyansh Sangule
3 years, 8 months ago

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I suspect you have that the answer is 13. This is incorrect, as the circles are not tangent to the lines at the same points. To fix this, simply do not give the radius of circle A. I will assume the radius of circle A is unknown and that the circles are tangent to the lines in the same points in the following.

Let the centers of the circles be \(O_A,O_B,O_C,O_D\). Let \(P\) be the intersection point of the tangents, and let circles B and C be tangent at \(M\). Let the radius of circle A be \(r_a\), and similar for circles B,C,D. We will attempt to find \(O_AO_C\).

Note that \(r_b=3\). Since \(A1,A2,A3\) are in arithmetic progression, \(m\angle A2=90^\circ\), and so \(m\angle O_AO_BO_C=90^\circ\).

By symmetry, \(P\) lies on \(O_AO_C\). Since \(\angle O_CMP\cong\angle O_CO_BO_A\) and \(\angle MO_CP\cong\angle O_BO_CO_A\), triangles \(\triangle O_CMP\) and \(\triangle O_CO_BO_A\) are similar by AA similarity. Therefore, \[\dfrac{r_c}{r_b}=\dfrac{r_c+r_b}{r_b+r_a}\implies r_a=1\]

Then, \(\triangle O_AO_BO_C\) is a right triangle, and \[O_AO_C=\sqrt{O_AO_B^2+O_BO_C^2}=\sqrt{(r_a+r_b)^2+(r_b+r_c)^2}=\sqrt{4^2+12^2}=4\sqrt{10}\] Daniel Chiu · 3 years, 8 months ago

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@Daniel Chiu Oh! Thank You for Pointing it out ! :) Nice Solution ;) Priyansh Sangule · 3 years, 7 months ago

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Are the adjacent circles supposed to be tangent to the line between them on the same point? Daniel Liu · 3 years, 8 months ago

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@Daniel Liu Oh yeah . Priyansh Sangule · 3 years, 8 months ago

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Calculate the radius of B to be 3 units and A2 would be 90 degrees

Get the distance to be \sqrt{13} + \sqrt{90} == 13.092 Sanjay Banerji · 3 years, 8 months ago

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@Sanjay Banerji Well , somethings not right Priyansh Sangule · 3 years, 8 months ago

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