Interesting geometry problem

[lev888]

From my son's Russian School of Mathematics homework.



The diagram contains an "extra" segment, which is a hint:

Spoiler: Diagram

 

[Dr.Peterson]

I solved it using a different auxiliary segment:

Spoiler

I suspect the segment in their diagram may require using a theorem I haven't memorized, for which my segment may be part of the proof.

 

[lev888]

I solved it using a different auxiliary segment:

Spoiler

View attachment 30954

I suspect the segment in their diagram may require using a theorem I haven't memorized, for which my segment may be part of the proof.

Interesting. Don't see yet where this is leading. My solution took advantage of the segment in the diagram and the relationships between areas of triangles with the same height. Is the theorem you mentioned relevant to those?

 

[Dr.Peterson]

Interesting. Don't see yet where this is leading. My solution took advantage of the segment in the diagram and the relationships between areas of triangles with the same height. Is the theorem you mentioned relevant to those?

I'm probably just missing something obvious in your method (which I often do in this sort of problem). My method uses proportionality of area to base and height, as well as similar triangles. It's probably far too complicated.

What I'm not using is some theorem that would directly tell me the relative lengths of the segments from P. (I can get those indirectly.)

 

[lev888]

My method uses proportionality of area to base and height, as well as similar triangles.

Same here.