Investigate the similarity of triangles

Considering ∠ b = ∠ c = 60.

From this perspective, let's discuss it first.

Then BD/BP=CE/CP.

Let BP = X.

At this time, 3/x=4/(8-x)

The solution is x=24/7.

Or BD/BP=CP/CE.

That is 3/x=(8-x)/4.

So x=2 or-10 (rounded)

If angles b and c don't agree.

Then b corresponds to another angle, both 60 degrees.

So two triangles are regular triangles,

At this time, BP = 3 and CP = 4.

inconnected

To sum up, BP=24/7 or 2.