Three Equal Distances

★ 8/10plane geometryCirclesanalytic geometryCasework

Problem

Let α,β,γ∈[0,2π)\alpha, \beta, \gamma \in [0, 2\pi) be pairwise distinct. Determine, with proof, which of the values 00, 11, 22, 44 can be the number of solutions (x,y)(x, y) of the system

∣xcos⁡α+ysin⁡α+1∣=∣xcos⁡β+ysin⁡β+1∣=∣xcos⁡γ+ysin⁡γ+1∣.|x\cos\alpha + y\sin\alpha + 1| = |x\cos\beta + y\sin\beta + 1| = |x\cos\gamma + y\sin\gamma + 1|.

Answer

Solution

Difficulty8/10
Topicsplane geometry, Circles, analytic geometry, Casework

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments
Log in to join the discussion.

No comments yet.