This is a excellent question.
I did not quite follow the originator's proof.
I forget the definition of directrix, and find it just from web site. Could you give the details of
由角平分线的定义和椭圆及准线的定义易证明三角形APC和APP'相似.
Your APC should be the same as CPR in my proof. However, I did not found the way to prove the similarity without calculation.
Futher question: do we need that P, C, Q on a line?
Another idea of the proof (mainly follow 乱弹)
First it is easy to get Q.
Definitely, there is a circle crossing P, C (focus) and Q. Only need to prove that the center of the circle is on Y aixs, then this circle automatically crosses another focus C'. The calculation should be not too hard. Somebody may be interested in giving details.
I did not quite follow the originator's proof.
I forget the definition of directrix, and find it just from web site. Could you give the details of
由角平分线的定义和椭圆及准线的定义易证明三角形APC和APP'相似.
Your APC should be the same as CPR in my proof. However, I did not found the way to prove the similarity without calculation.
Futher question: do we need that P, C, Q on a line?
Another idea of the proof (mainly follow 乱弹)
First it is easy to get Q.
Definitely, there is a circle crossing P, C (focus) and Q. Only need to prove that the center of the circle is on Y aixs, then this circle automatically crosses another focus C'. The calculation should be not too hard. Somebody may be interested in giving details.
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