AoPS - Problem

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Problem

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Tags: parallel, geometry, circles

parmenides51

25.03.2021 17:55

The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$ , so that point $O$ lies on circle $C_2$ . The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$ , prove that the lines $BD$ and $d$ are parallel.

donian9265

25.03.2021 18:17

Let the center of

$C_2$ be

$P$ . Then let

$\angle OAD=\angle ODA=\alpha$ . Therefore,

$\angle OAP=\angle OBP=90-\alpha$ and

$\angle AOD=189-2\alpha$ . Now extend

$d$ until it intersects

$C_2$ at

$E$ . Notice that

$\angle DAE=90+\alpha$ so to prove

$BD$ and

$d$ are parallel, it suffices to prove that

$\angle ADB=90-\alpha$ . We will now let

$\angle AEB=\beta$ so

$\angle APB=2\beta$ and

$\angle AOB=180-\beta$ . Now look at quadrilateral

$AOBP$ . Since the angles in this add to

$360$ ,

$180+\beta=180+2\alpha$ so

$\beta=2\alpha$ . Now look at the angles around

$O$ . Since these must add to

$360$ ,

$\angle DOB=4\alpha$ so

$\angle ODB=90-2\alpha$ . Thus,

$\angle ADB=90-\alpha$ as desired.