RMO 2000 Problem 1 - Learn about Cyclic Pentagon configuration in Geometry

Let 𝐴𝐶 be a line segment in the plane and 𝐵 a point between 𝐴 and 𝐶. Construct isosceles triangles 𝑃𝐴𝐵 and 𝑄𝐵𝐶 on one side of the segment 𝐴𝐶 such that ∠𝐴𝑃𝐵=∠𝐵𝑄C=120∘ and an isosceles △𝑅𝐴𝐶 on the other side of 𝐴𝐶 such that ∠𝐴𝑅𝐶=120∘. Show that 𝑃𝑄𝑅 is an equilateral triangle.

Problem useful for I.S.I B.Stat B.Math Entrance, CMI Entrance and Math Olympiad

[deepjyoti]

If anyone proof only <DFE=60
Then also done
Cause it will lead to
<ADF+<FEB=180 degrees
Now in triangle
ADF and FEB
We have AF=BF, <FAD=<FEB, and
<ADF+<FEB=180
This will lead to DF=FE
Now <DFE=60 and DF=FE

[anomitrosikdar]

angle CNB = 120° which means that angle CNR = 60°. Since points P, R, N, and Q are cyclic it implies that angle PNR = angle PQR = 60°

(angle CNR = angle PNR because points P, C, N are collinear)