Simple(?) proof on points on bisecting lines.

[alex909]

Hey all,


I'm an adult going back through geometry (the McDougal text). Only in chapter 4, but it's going smoothly. However, there's one particular problem that is tripping me up - it seems like I'm supposed to assume something that isn't necessarily true. Either that or I'm missing the obvious, but I've asked some other people and they don't see anything either.


Any help would be very appreciated .

 

[stapel]

...there's one particular problem that is tripping me up - it seems like I'm supposed to assume something that isn't necessarily true. Either that or I'm missing the obvious, but I've asked some other people and they don't see anything either.

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\(\displaystyle \mbox{Let }\, \angle\, ABC\, \mbox{ be an angle. Let }\, D\, \mbox{ and }\, E\, \mbox{ be points lying on }\, \overrightarrow{BA}\, \mbox{ and }\, \overrightarrow{BC},\,\)

\(\displaystyle \mbox{respectively, with }\, D\, \mbox{ between }\, B\, \mbox{ and }\, A,\, \mbox{ and with }\, E\, \mbox{ between }\, B\, \mbox{ and }\, C.\)

\(\displaystyle \mbox{ Let }\, P\, \mbox{ be a point inside }\, \angle\, ABC\, \mbox{ such that:}\)

. . . . .\(\displaystyle \mbox{i. }\, \overrightarrow{DP}\, \mbox{ bisects }\, \angle\, ADE\)

. . . . .\(\displaystyle \mbox{ii. }\, \overrightarrow{EP}\, \mbox{ bisects }\, \angle\, DEC\)

\(\displaystyle \mbox{Prove that }\, \overrightarrow{BP}\, \mbox{ bisects }\, \angle\, ABC\)

What is the "assumption that isn't necessarily true" that you think you are needing to make? Thank you!

 

[alex909]

Honestly, I wasn't even quite sure about that - perhaps that BD and BE are congruent, or that the intersection of BP and DB is a midpoint and/or perpendicular . . . that DP is congruent to EP?

Does your question mean that you do know how to solve it? I partially just want to make sure it's not a typo and that there is a solution before I keep banging my head against it .

Thanks!