How to prove it?

Feb 2010
738
162
Which angle is 30 degrees? You state one at the top of the picture but mark a different one in the diagram.
 

skipjack

Forum Staff
Dec 2006
21,481
2,470
This is quite easy to do using trigonometry, which also establishes that angles BAP and PAC are respectively 24° and 12°. Was your trigonometry question related to this problem?

I haven't found an elegant geometrical proof yet. Where does this problem come from?
 
Feb 2020
10
0
shanghai
This is quite easy to do using trigonometry, which also establishes that angles BAP and PAC are respectively 24° and 12°. Was your trigonometry question related to this problem?

I haven't found an elegant geometrical proof yet. Where does this problem come from?
Yes, my trigonometry question is from this problem. This problem is from a Chinese student. I guess it's a competition problem.
 

skipjack

Forum Staff
Dec 2006
21,481
2,470
Construct the regular pentagon $AEBCD$. Angle $BAC = 36^\circ$.
Construct the perpendicular $CH$ from $C$ to $AE$.
Construct the equilateral triangle $AEQ$. By symmetry, $Q$ lies on $CH$.
Draw the line segment $BQ$.
The angles shown below are easily calculated.
Hence $Q$ coincides with the point $P$ given in the problem and so $AP$ = $BC$.
Pentagon.JPG
 
Mar 2017
33
0
Norway
Construct the regular pentagon $AEBCD$. Angle $BAC = 60^\circ$.
Construct the perpendicular $CH$ from $C$ to $AE$.
Construct the equilateral triangle $AEQ$. By symmetry, $Q$ lies on $CH$.
Draw the line segment $BQ$.
The angles shown below are easily calculated.
Hence $Q$ coincides with the point $P$ given in the problem and so $AP$ = $BC$.
View attachment 11005
I think there is a typo here, it should be $BAC = 36^\circ$
 

skipjack

Forum Staff
Dec 2006
21,481
2,470
Thanks. I've corrected it.