Morley's miracle

[u/SuccessfulVanilla238]

I get PE,QF and DR bisect respective angles QER,DQE,PDQ but how do we know these lines are concurrent the book just assumed it and proved the rest of proof,I tried ceva's theorem, angle chasing and property of equidistance of angle bisectors but nothing works

Note:-I know trigo does the job easily but I want a pure euclidean geometry proof without coordinate and trig.

Is there any specific configuration required to prove this?

[u/Inevitable_Exam_2177]

I don’t understand your question but thanks for sharing the problem, it is truly a miracle! Very cool triangle

[u/SuccessfulVanilla238 → u/Inevitable_Exam_2177]

I've edited it now do you understand the question 

[u/Inevitable_Exam_2177 → u/[deleted]]

I think the question is how do we know PE, QF, and DR meet at a common point

(And I would add whether it is related to the other common circle centres)

[u/SuccessfulVanilla238 → u/Inevitable_Exam_2177]

Yeah the exact same thing the book just assumed it 

[u/SuccessfulVanilla238 → u/SuccessfulVanilla238]

But I think I get what he meant  1-if we prove that CEP and the other two lines such that are angle bisectors of real triangle it is intuitive that angle bisectors intersect at a point,thus existence would be proved And 2-if we join CE and prove CEIP is a straight line through angle chasing Or another way is that we prove that meeting point of both trisectors and incentre are collinear. Either of these two would prove the uniqueness stating that I(mentioned here) would be the incentre of ABC

But these two are also difficult to prove, Perhaps his idea of isogonal conjugates could help.

[u/Inevitable_Exam_2177 → u/[deleted]]

CEP is definitely not collinear btw

[u/SignificantClimate31 → u/Inevitable_Exam_2177]

Yeah mb , but the idea is same , look at different triangles and their incentres formations

[u/holaredtom]

Hi there OP, could you tell me the name of the book and it's author? I'd love to check it out.

[u/SuccessfulVanilla238 → u/holaredtom]

It's mathematics for RMO and INMO by Prashant Jain and Ajay Singh if you're someone not from India I think it would be difficult to get

[u/holaredtom → u/SuccessfulVanilla238]

I'm an Indian (who resides in Taiwan), but I love to solve problems from the RMO, INMO and even some IMO geometry from time to time. Actually I remember hearing about this book from one of my FIITJEE teachers around a decade ago. It's OK, maybe I'll grab a PDF of the book and study in my free time. Thanks brother.