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In the solution of this question, why do all 3 points :- the vertex B, tangential point of quarter circle and external circle, center of circle need to be collinear ?

In my rough diagram, they are not being collinear, how can I prove these 3 points to be collinear?

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  • $\begingroup$ Redraw your diagram more accurately. In a squeezed diagram both circles become ellipses. $\endgroup$
    – Intelligenti pauca
    Commented May 13, 2023 at 7:45
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    $\begingroup$ You should use Geogebra for example to draw your diagrams... $\endgroup$
    – Jean Marie
    Commented May 13, 2023 at 13:44

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As noted in a comment, using a sketch that you yourself call a “rough diagram” is leading you astray. However, even with an accurate diagram, the question remains valid: why are the centres of the two circles collinear with the point where they touch?

When we say that two circles “touch externally”, we mean that they have a common tangent line at that point, with their centres on opposite sides of this line. A radius of a circle is always perpendicular to the tangent line it meets on the circumference. (Euclid III.18)

A diagram of the problem. The rectangle, large quarter circle, and small full circle are shown with solid lines. Their shared tangent line, and several radiuses, are shown with dashed lines.

Call the small circle’s centre $O$, and the touch point $T$. The two circles’ radiuses $BT$ and $OT$ are both perpendicular to a third line, their common tangent, so $BT \parallel OT$. And they share point $T$, so they are collinear. ∎

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  • $\begingroup$ Thank you, how did you get BT || OT , is it possible for you to make a diagram and explain, I am not really getting how did you prove the collinearity in the last part of your answer ? $\endgroup$
    – Fin27
    Commented May 13, 2023 at 9:27
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    $\begingroup$ @Fin27 I’ll try and put a diagram together when I have a chance, but in words: Two lines that are both perpendicular to a third must be parallel to each other. (Two posts set upright on the same floor must be parallel.) And parallel lines that share even one point are, in fact, the “same” line—they are collinear. $\endgroup$
    – Tim Pederick
    Commented May 14, 2023 at 4:39
  • $\begingroup$ Understood, Thanks a lot :)... I have one more extension to my doubt , will this line from vertex B passing through the center of smaller circle also touch the opposite vertex D as well ? $\endgroup$
    – Fin27
    Commented May 15, 2023 at 10:46
  • $\begingroup$ @Fin27 (Oops, thought I replied to this already…) No. D and O are opposite vertices of a square with two radiuses of the small circle as edges. So its diagonal DO is at 45° to DC and DA. But rectangle ABCD isn’t square, so its diagonal BD is not at 45° to DC or DA. $\endgroup$
    – Tim Pederick
    Commented May 18, 2023 at 9:26
  • $\begingroup$ Thank you so much :) $\endgroup$
    – Fin27
    Commented May 19, 2023 at 20:38

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