PROOF 152

Prove.

A tangent to a circle is perpendicular to the radius

On the Euclidean plane.

What you may use

A1circle = all points P with OP = r
definition of the circle
A2through the center ⟹ meets at two points
diameter — a line through the center meets the circle at two distinct points
A3the foot of the perpendicular exists uniquely
perpendicular from a point off a line
A4the perpendicular makes right angles with points of ℓ
perpendicularity — the angle at the foot with any other point of the line is right
A5a point splits a line into two rays
separation of a line
A6a point at a given length on a ray
transferring a segment (Euclid I.3)
A7two sides and the included angle ⟹ congruent
SAS congruence condition (Euclid I.4)
A8congruent ⟹ corresponding sides equal
definition of congruence
A9in a right triangle p²+q²=c²
Pythagorean theorem (Proof 56)
A10u > 0 ⟹ u² > 0, v+u > v
order axioms
A11u,v≥0, u²>v² ⟹ u>v
monotonicity of the square root
A12PQ ≥ 0, PQ = 0 ⟺ P = Q
properties of length
Rassumption contradictory ⟹ negation true
proof by contradiction (Proof 26)

Nothing outside this list may be cited.

Definition D1

The circle with center O and radius r is the set of all points P with OP = r. A line ℓ meeting this circle at exactly one point is called a tangent, and that one point the point of tangency.

Proof.

Related labs

Circles lab →