circle as a locus with radius <= radius of another fixed circle, equality case

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circle as a locus with radius <= radius of another fixed circle, equality case

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Source: Australian MO 2015

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#1 h Sep 24, 2018, 3:16 PM • 1 Y

Y by Adventure10

Let $\Gamma$ be a fixed circle with centre $O$ and radius $r$ . Let $B$ and $C$ be distinct fixed points on $\Gamma$ . Let $A$ be a variable point on $\Gamma$ , distinct from $B$ and $C$ . Let $P$ be the point such that the midpoint of $OP$ is $A$ . The line through $O$ parallel to $AB$ intersects the line through $P$ parallel to $AC$ at the point $D$ .
(a) Prove that, as $A$ varies over the points of the circle $\Gamma$ (other than $B$ or $C$ ), $D$ lies on a fixed circle whose radius is greater than or equal to $r$ .
(b) Prove that equality occurs in part (a) if and only if $BC$ is a diameter of $\Gamma$ .

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#2 h Jun 15, 2019, 6:53 PM • 1 Y

Y by Adventure10

Circle with equation $x^{2}+y^{2}=r^{2}$ .

Variable point $A(r\cos \alpha,r\sin \alpha)$ , fixed points $B(r,0),C(r\cos \beta,r\sin \beta)$ .

Point $P(2r\cos \alpha,2r\sin \alpha)$ .

Locus of the points $D$ is the circle $x^{2}+y^{2}+2rx-2r\cot \frac{\beta}{2}y=0$ .

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