Make geometric definitions testable

Try

Draw a dot marked O and a loop around it. Does that drawing define a circle? A rounded loop can have points at different distances from O. A definition must let another person decide membership without trusting the artist. For a concrete test, take O=(1,2) and radius 5. The point (4,6) belongs because its horizontal and vertical differences are 3 and 4, and 3²+4²=25. The point O itself does not belong to this circle.

Here and throughout this unit we work in the Euclidean plane. Points, lines and incidence are foundational notions, not objects defined by circular phrases such as “a line is a set of collinear points.” Coordinates provide a model in which we can test derived definitions. This is a school reasoning unit, not a complete axiomatization of geometry.

Predict

Before calculating, classify (6,2), (1,7), (1,2) and (5,5) relative to that circle. Predict which answers would change if “distance exactly 5” became “distance at most 5.” Then draw two short segments that do not touch, but whose supporting full lines would meet if extended. The second action tests whether “never touch in this drawing” can define parallel lines.

Experiment

Complete this membership table on paper before reading its final column. Squared distance avoids unnecessary square roots; it is valid here because distance and radius are nonnegative.

Point	Squared distance from (1,2)	Circle membership
(6,2)	5²+0²=25	Yes
(1,7)	0²+5²=25	Yes
(1,2)	0	No; inside the disk
(5,5)	4²+3²=25	Yes

Now place A=(0,0) and B=(2,0). Mark one point on segment AB, one on ray AB but outside the segment, and one on the full line AB but outside the ray. For example, (1,0), (3,0) and (−1,0) meet these three requests. Changing the second endpoint from B to A would destroy the direction needed to name ray AB: the two defining points must be distinct.

Understand

A segment AB consists of A, B and the points between them. In coordinates its points are A+t(B−A) with 0≤t≤1. A ray AB uses t≥0; its endpoint is A. A full line uses every real t. Here t is a dimensionless position parameter, not a length. Distance between points is the length of their joining segment. The coordinate formula comes from an independently established right-triangle Pythagorean theorem; later similarity arguments will not be used to justify their own premises.

An angle is determined by two rays with a common endpoint, together with the selected region when ambiguity matters. A right angle measures 90 degrees. Two intersecting lines are perpendicular when their adjacent angles are right angles. Two parallel lines are distinct, coplanar full lines with no common point. “Coplanar” matters in space: skew lines can be nonintersecting without being parallel. Coincident lines are not distinct parallel lines under this convention.

A circle of radius r>0 and center O is the set of points at distance exactly r from O. The closed disk uses distance at most r. An arc is a connected portion of the circle between specified endpoints. Usually two arcs join distinct endpoints; specify the minor or major arc, or a point through which it passes. Antipodal endpoints yield two semicircles of equal length, so “the shorter arc” fails to select one.

Apply

Repair the claim “perpendicular lines look upright and horizontal.” Rotate the paper: the right angle remains, but those visual directions change. Repair “a circle contains its center” by deciding whether the speaker meant a disk. For each repair, supply an exact condition and a counterexample to the original wording. Numerical examples can refute an overly broad definition; they cannot prove that every object satisfying a corrected definition has some later theorem property.

A good written answer identifies the objects, states the condition, and shows why the example passes or fails it. Neither a rough drawing nor a list of vocabulary substitutes for those decisions.

Continue

Next, treat motions as functions on every point of the plane. Keep the distinctions between a point, a segment and a full line: a rule tested at two or three sample points is not automatically a rule about an entire figure. The practice below asks you to make these distinctions explicit with fresh coordinates.