Circles Keep Distances

Part of the Transformations & Constructions section of Coddy's Geometry journey. Lesson 37 of 61.

A construction uses two tools and no ruler marks. The straightedge draws the line through two points. The compass draws the circle with a center through a point. New points appear only where lines and circles cross. Nothing is measured, so the result is exact.

$Circles about P through Q and about Q through P, crossing at R and S$

The key fact: every point on the circle about P through Q is exactly PQ away from P. So a point on both circles, the one about P through Q and the one about Q through P, is PQ from P and PQ from Q. The circles cross at R and S, and each makes a triangle with P and Q whose three sides are all equal to PQ.

On the board, pick Circle and tap the center, then the point it passes through. Pick Point and tap where two circles cross to mark it; the board names the points you make P1, P2 and so on. The counter shows the tools you used and the best possible.

Challenge

Construct the point C that makes triangle ABC equilateral: C must be exactly AB from A and exactly AB from B. Draw two circles, mark a crossing with Point, then press Check.

Try it yourself

$Construct it with a compass and straightedge$