Give a geometrical construction for finding the fourth point lying on a circle passing through three given points, without finding the center of the circle. Justify the construction.

Question

Philoid · Accepted Answer

Let A, B, C, D be the given points.

With B as center and radius equal to AC draw an arc.

With C as center and AB as radius draw another arc.

Which cuts the previous arc at D,

Then, D is the required point BD and CD.

In ΔABC and ΔDCB,

AB = DC

AC = DB

BC = CB

∴ ΔEBC ≅ ΔEDA

⇒ ∠BAC = ∠CDB

Thus, BC subtends equal angles, ∠BAC and ∠CDB on the same side of it.

Therefore, points A, B, C, D are concyclic.