concyclic due to equal angles and perpendiculars (Estonia Open 2000 J 2.4)

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concyclic due to equal angles and perpendiculars (Estonia Open 2000 J 2.4)

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#1 h Apr 7, 2020, 2:09 AM • 1 Y

Y by Mango247

In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$ . From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C the perpendicular on $OB$ intersects the ray $OD$ at point $H$ . Prove that the points $C, D, G$ and $H$ are conlyclic.

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#2 h Apr 7, 2020, 12:26 PM • 1 Y

Y by gabrielghx

Let $DG \cap AO$ at $P$ and $CH \cap BO$ at $Q$
Note that $\angle CGD =\angle PGD=180^\circ-\angle OAG-\angle AOC=180^\circ-\angle HQO-\angle HOQ=\angle OHQ=\angle CHD$
Therefore, $C,D,G,H$ are concyclic.

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