Geometry_5

Diagram:

Solvable with angle chasing
(Will post complete solutions to Geo #5 and Geo #3 next week)
(Also, the angles are not necessarily 23.33 - my construction just makes them 23.33)

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Eh, might as well just finish this one:
Solution:


Let’s first set \angle QAR=2x, \angle QOR=2y, \angle COQ=2q and \angle ABC =2z (the goal is to now arrive to the conclusion that x=z.

From here we can derive a plethora of angles:
\angle PBR=2x (same arc with respect to big circle)
\angle CBR=q+y (inscribed angle of small circle)
\angle CBP = q+y-2x (simple subtraction)
\angle ARP=y+q (inscribed tangent theorem - perhaps I have the wrong name)
[it states that \angle ARP is equal to half the angle of arc CR]

Now we know that APRB is cyclic.
Thus \angle ABP=\angle ARP,
This means that 2z+q+y-2x=y+q.
This simplifies to z=x and we are done!

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:laughing: :laughing: :laughing: :laughing: ‘I have seen the leaderboard and will be finishing this question now’

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XD

How do you make those diagrams ?

I asked @AwesomeLife_Math the same thing!

Apparently it’s called Geogebra :slightly_smiling_face:

2019_Question_2

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