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In Fig.1 given that DE, DF & DG are respective perpendiculars to AB, AC & BC

Prove that E, F & G are collinear.

(I found this problem on youtube but on trying to solve through co-ordinate geometry, the expression becomes very complex after a few steps, is there any elegant solution!)


In Fig.2 given that EC & ED are tangents to the semicircle

Prove that EO is perpendicular to AB.

(I found this problem on twitter and was able to solve it through co-ordinate geometry but the steps were too long and complex. There should be an elegant solution)

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    $\begingroup$ See Simson line. $\endgroup$ – Saad Jun 13 '18 at 15:24
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Fig. 1 is just related to the classical Simson line, whose existence depends on angle chasing.
About Fig. 2, the hint I give you is to consider the intersection of $AC$ and $BD$.
This point $Q$ is collinear with $E$ and $O$ and it lies on the circle through $C,O,D$.

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Since $O$ is the orthocenter of $ABQ$ the claim $EO\perp AB$ is a straightforward consequence.
You may also notice that the circle through $C,D,E$ is the nine-point circle of $ABQ$.

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