According to the concept of equilateral points, we can know that triangles with internal angles of 30, 60 and 90 have equilateral points, but equilateral triangles do not, so we can judge them accordingly.
According to △ABC, ∠BPC=∠ABP+∠BAC+∠ACP and ∠BAC=∠PBC, the quantitative relationship between ∠BPC, ∠ABC and ∠ACP can be obtained.
First, connect PB and PC. Then, according to the fact that the intersection point P of the bisectors of the three internal angles of △ABC is the equilateral point of the triangle, and the sum of the internal angles of the triangle is 180, the equation about ∠A is obtained, and the degree of ∠A is obtained.
Equiangular characteristics:
Let e and f be two points in ∠APB and satisfy ∠APF=∠BPE. Let e and f be axisymmetrical points s and T:FS = et about PA and PB. “FS=ET。” The equivalent expression of Pang Si's theorem is: "A pair of equiangular points E and F in ∠APB satisfy ∠APF=∠BPE.
Another necessary and sufficient condition that the reflection paths on both sides of PA and PB are equal in length is that both sides are four projective circles! The center of the circle is the midpoint of this set of equiangular points.