# How to prove it?

How to prove it?

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#### mrtwhs

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

#### Ricardo_2020

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

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#### skipjack

Forum Staff
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?

#### Ricardo_2020

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
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 $AP$ = $BC$.

#### Farzin

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
Thanks. I've corrected it.