3 common tangents of 2 non intersecting circles (2014 Belgium OMB Juniors p1)

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3 common tangents of 2 non intersecting circles (2014 Belgium OMB Juniors p1)

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#1 h Dec 20, 2020, 3:29 AM

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The lines $d_1$ and $d_2$ are tangent on the outside to two circles with distinct radii, while $d$ their is tangent on the inside. The line $d$ is tangent to the circles at $A$ and $B$ , the line $d_1$ at $A_1$ and $B_1$ and the line $d_2$ at $A_2$ and $B_2$ . The line $d$ cuts $d_1$ at $X$ and $d_2$ at $Y$ .
(a) Suppose first that, further, the two circles are tangent to each other.
(i) Adapt the figure to this particular case.
(ii) Then express $|A_1B_1|$ in terms of the radii of the two circles.
(b) In the general case, prove that
(i) $|A_1B_1| =|A_2B_2|$
(ii) $|A_1B_1| =|XY|$
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This post has been edited 2 times. Last edited by parmenides51, Dec 20, 2020, 3:41 AM

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